Pfaffian Schur Processes and Last Passage Percolation in a Half-Quadrant1

The Annals of Probability 2018, Vol. 46, No. 6, 3015–3089 https://doi.org/10.1214/17-AOP1226 © Institute of Mathematical Statistics, 2018

PFAFFIAN SCHUR PROCESSES AND LAST PASSAGE PERCOLATION IN A HALF-QUADRANT1

∗ BY JINHO BAIK ,2,GUILLAUME BARRAQUAND†,3,IVA N CORWIN†,4 AND TOUFIC SUIDAN University of Michigan∗ and Columbia University†

We study last passage percolation in a half-quadrant, which we analyze within the framework of Pfaffian Schur processes. For the model with exponential weights, we prove that the fluctuations of the last passage time to a point on the diagonal are either GSE Tracy–Widom distributed, GOE Tracy– Widom distributed or Gaussian, depending on the size of weights along the diagonal. Away from the diagonal, the fluctuations of passage times follow the GUE Tracy–Widom distribution. We also obtain a two-dimensional crossover between the GUE, GOE and GSE distribution by studying the multipoint distribution of last passage times close to the diagonal when the size of the diagonal weights is simultaneously scaled close to the critical point. We expect that this crossover arises universally in KPZ growth models in half-space. Along the way, we introduce a method to deal with diverging correlation kernels of point processes where points collide in the scaling limit.

CONTENTS 1. Introduction ...... 3016 1.1. Previous results in (half-space) LPP ...... 3017 1.2. Previous methods ...... 3018 1.3. Our methods and new results ...... 3018 1.4.OthermodelsrelatedtoKPZgrowthinahalf-space...... 3020 1.5.Mainresults...... 3020 Outline of the paper ...... 3024 2. Fredholm Pfafﬁan formulas ...... 3024 2.1.GUETracy–Widomdistribution...... 3024 2.2. Fredholm Pfafﬁan ...... 3025 2.3.GOETracy–Widomdistribution...... 3026 2.4.GSETracy–Widomdistribution...... 3027

Received July 2016; revised August 2017. 1Supported by NSF PHY11-25915. 2Supported in part by NSF Grant DMS-1361782. 3Supported in part by the LPMA UMR CNRS 7599, Université Paris-Diderot–Paris 7 and the Packard Foundation through I. Corwin’s Packard Fellowship. 4Supported in part by the NSF through DMS-1208998, a Clay Research Fellowship, the Poincaré Chair and a Packard Fellowship for Science and Engineering. MSC2010 subject classiﬁcations. Primary 60K35, 82C23; secondary 60G55, 05E05, 60B20. Key words and phrases. Last passage percolation, KPZ universality class, Tracy–Widom distributions, Schur process, Fredholm Pfafﬁan, phase transition. 3015 3016 BAIK, BARRAQUAND, CORWIN AND SUIDAN

2.5.Crossoverkernels...... 3029 3.PfaffianSchurprocess...... 3031 3.1. Definition of the Pfaffian Schur process ...... 3031 3.1.1. Partitions and Schur positive specializations ...... 3031 3.1.2. Useful identities ...... 3033 3.1.3. Definition of the Pfaffian Schur process ...... 3033 3.2.PfaffianSchurprocessesindexedbyzig-zagpaths...... 3036 3.3.DynamicsonPfaffianSchurprocesses...... 3037 3.4.Thefirstcoordinatemarginal:Lastpassagepercolation...... 3041 4. Fredholm Pfaffian formulas for k-pointdistributions...... 3044 4.1.Pfaffianpointprocesses...... 3045 4.2.Half-spacegeometricweightLPP...... 3046 4.3.Deformationofcontours...... 3048 4.4. Half-space exponential weight LPP ...... 3050 5.ConvergencetotheGSETracy–Widomdistribution...... 3058 5.1.AformalrepresentationoftheGSEdistribution...... 3059 5.2. Asymptotic analysis of the kernel Kexp ...... 3068 5.3.Conclusion...... 3073 6.Asymptoticanalysisinotherregimes:GUE,GOE,crossovers...... 3074 6.1. Phase transition ...... 3074 6.2.GOEasymptotics...... 3076 6.3.GUEasymptotics...... 3078 6.4.Gaussianasymptotics...... 3084 6.5. Convergence of the symplectic-unitary transition to the GSE distribution ...... 3086 Acknowledgments...... 3087 References...... 3087

1. Introduction. In this paper, we study last passage percolation in a half- quadrant of Z2. We extend all known results for the case of geometric weights (see discussion of previous results below) to the case of exponential weights. We study in a unified framework the distribution of passage times on and off diagonal, and for arbitrary boundary condition. We do so by realizing the joint distribution of last passage percolation times as a marginal of Pfaffian Schur processes. This allows us to use powerful methods of Pfaffian point processes to prove limit theorems. Along the way, we discuss an issue arising when a simple point process converges to a point process where every point has multiplicity two, which we expect to have independent interest. These results also have consequences about interacting particle systems—in particular the TASEP on positive integers and the facilitated TASEP—that we discuss in [2].

DEFINITION 1.1 (Half-space exponential weight LPP). Let (wn,m)n≥m≥0 be a sequence of independent exponential random variables5 with rate 1 when n ≥ m +

5The exponential distribution with rate α ∈ (0, +∞), denoted E(α), is the probability distribution −αx on R>0 such that if X ∼ E(α), P(X > x) = e . PFAFFIAN SCHUR PROCESSES AND LPP IN A HALF-QUADRANT 3017

FIG.1. LPP on the half-quadrant. One admissible path from (1, 1) to (n, m) is shown in dark gray. H(n,m) is the maximum over such paths of the sum of the weights wij along the path.

1 and with rate α when n = m. We deﬁne the exponential last passage percolation (LPP) time on the half-quadrant, denoted H(n,m), by the recurrence for n ≥ m, max H(n− 1,m),H(n,m− 1) if n ≥ m + 1, H(n,m)= wn,m + H(n,m− 1) if n = m with the boundary condition H(n,0) = 0.

It is useful to notice that the model is equivalent to a model of last passage percolation on the full quadrant where the weights are symmetric with respect to the diagonal (wij = wji).

REMARK 1.2. If one imagines that the light gray area in Figure 1 corresponds to the percolation cluster at some time, its border (shown in black in Figure 1) evolves as the height function in the Totally Asymmetric Simple Exclusion process on the positive integers with open boundary condition, so that all our results could be equivalently phrased in terms of the latter model.

1.1. Previous results in (half-space) LPP. The history of fluctuation results in last passage percolation goes back to the solution to Ulam’s problem about the asymptotic distribution of the size of the longest increasing subsequence in a uniformly random permutation [4]. A proof of the nature of fluctuations in exponential distribution LPP is provided in [29]. On a half-space, [5–7] studies the longest increasing subsequence in random involutions. This is equivalent to a half-space LPP problem since for an involution σ ∈ Sn, the graph of i → σ(i) is symmetric with respect to the first diagonal. Actually, [5–7] treat both the longest increasing subsequences of a random involution and symmetrized LPP with geometric weights. That work contains the geometric weight half-space LPP analogue of Theorem 1.3. 3018 BAIK, BARRAQUAND, CORWIN AND SUIDAN

The methods used therein only worked when restricted to the one point distribution of passage times exactly along the diagonal. Further work on half-space LPP was undertaken in [38], where the results are stated there in terms of the discrete polynuclear growth (PNG) model rather than the equivalent half-space LPP with geometric weights. The framework of [38] allows to study the correlation kernel corresponding to multiple points in the space direction, with or without nucleations at the origin. In the large scale limit, the authors performed a nonrigorous critical point derivation of the limiting correlation kernel in certain regimes and found Airy-type process and various limiting crossover kernels near the origin, much in the same spirit as our results (see the discussion about these results in Section 1.3).

1.2. Previous methods. A key property in the solution of Ulam’s problem in [4] was the relation to the RSK algorithm which reveals a beautiful algebraic structure that can be generalized using the formalism of Schur measures [34] and Schur processes [35]. RSK is just one of a variety of Markov dynamics on sequences of partitions which preserves the Schur process [13, 17]. Studying these dynamics gives rise to a number of interesting probabilistic models related to Schur processes. In the early works of [4, 6], the convergence to the Tracy–Widom distributions was proved by Riemann–Hilbert problem asymptotic analysis. Subsequently, Fredholm determinants became the preferred vehicle for asymptotic analysis. In a half-space, [5] applied RSK to symmetric matrices. Subsequently, [37](see also [26]) showed that geometric weight half-space LPP is a Pfaffian point process (see Section 4.1) and computed the correlation kernel. Later, [21] defined Pfaffian Schur processes generalizing the probability measures defined in [37]and[38]. The Pfaffian Schur process is an analogue of the Schur process for symmetrized problems.

1.3. Our methods and new results. We consider Markov dynamics similar to those of [17] and show that they preserve Pfaffian Schur processes. This enables us to show how half-space LPP with geometric weights is a marginal of the Pfaffian Schur process. We then apply a general formula giving the correlation kernel of Pfaffian Schur processes [21] to study the model’s multipoint distributions. We degenerate our model and formulas to the case of exponential weight half-space LPP (previous works [5–7, 38] only considered geometric weights). We perform an asymptotic analysis of the Fredholm Pfaffians characterizing the distribution of passage times in this model, which leads to Theorems 1.3 and 1.4.Wealso study the k-point distribution of passage times at a distance O(n2/3) away from the diagonal, and introduce a new two-parameter family of crossover distributions when the rate of the weights on the diagonal are simultaneously scaled close to their critical value. This generalizes the crossover distributions found in [25, 38]. The asymptotics of fluctuations away from the diagonal were already studied in [38] for the PNG model on a half-space (equivalently the geometric weight half- space LPP). The asymptotic analysis there was nonrigorous, at the level of studying PFAFFIAN SCHUR PROCESSES AND LPP IN A HALF-QUADRANT 3019

the behavior of integrals around their critical points without controlling the decay of tails of integrals. The proof of the first part of Theorem 1.3 (in Section 5) required a new idea which we believe is the most novel technical contribution of this paper: When α>1/2, H(n, n) is the maximum of a simple Pfaffian point process which converges as n →∞to a point process where all points have multiplicity 2. The limit of the correlation kernel does not exist6 as a function (it does have a formal limit involving the derivative of the Dirac delta function). Nevertheless, a careful reordering and nontrivial cancellation of terms in the Fredholm Pfaffian expansions allows us to study the law of the maximum of this limiting point process, and ultimately find that it corresponds to the GSE Tracy–Widom distribution. We actually provide a general scheme for how this works for random matrix type kernels in which simple Pfaffian point processes limit to doubled ones (see Section 5.1). Theorem 5.3 of [38] derives a formula for certain crossover distributions in half- space geometric LPP. This crossover distribution, introduced in [25], depends on a parameter τ (which is denoted η in our Theorem 1.8) and it corresponds to a natural transition ensemble between GSE when τ → 0 and GUE when τ →+∞.Al- though the collision of eigenvalues should occur in the GSE limit of this crossover, this point was left unaddressed in previous literature and the convergence of the crossover kernel to the GSE kernel was not proved in [25] (see our Proposi- tion 6.12). The formulas for limiting kernels, given by [25], (4.41), (4.44), (4.47) or [38], (5.37)–(5.40), make sense so long as τ>0. Notice a difference of integration domain between [25], (4.44), and [38], (5.39). Our Theorem 1.8 is the analogue of [38], Theorem 5.3, for half-space exponential LPP and we recover the exact same kernel. When τ = 0, the formula for I4 in [38], (5.39), involves divergent inte- 7 grals, though if one uses the formal identity R Ai(x + r)Ai(y + r)dr = δ(x − y) then it is possible to rewrite the kernel in terms of an expression involving the derivative of the Dirac delta function, and the expression formally matches with the kernel K∞ introduced in Section 5.1. From that (formal) kernel, it is nontrivial to match the Fredholm Pfaffian with a known formula for the GSE distribution, this matching does not seem to have been made previously in the literature and we explain it in Section 5.1. In random matrix theory, a similar phenomenon of collisions of eigenvalues occurs in the limit from the discrete symplectic ensemble to the GSE [22]. However, [22] used averages of characteristic polynomials to characterize the point process, and computed the correlation kernels from the characteristic polynomials only after the limit, so that collisions of eigenvalues were not an issue. A random matrix ensemble which crosses over between GSE and GUE was studied in [25].

6For a nonsimple point process, the correlation functions—Radon–Nikodym derivatives of the factorial moment measures—generically do not exist (see Deﬁnitions in Section 4.1). 7The saddle-point analysis in the proof of Theorem 5.3 in [38] is valid only when τ>0. Indeed, [38], (5.44), requires τ1 + τ2 >η1 + η2 and one needs that η1,η2 > 0 as in the proof of [38], Theorem 4.2. 3020 BAIK, BARRAQUAND, CORWIN AND SUIDAN

1.4. Other models related to KPZ growth in a half-space. A positive temperature analogue of LPP—directed random polymers—has also been studied. The log-gamma polymer in a half-space (which converges to half-space LPP with exponential weights in the zero temperature limit) is considered in [33]wherean exact formula is derived for the distribution of the partition function in terms of Whittaker functions. Some of these formulas are not yet proved and presently there has not been a successful asymptotic analysis preformed of them. Within physics, [28]and[14] studies the continuum directed random polymer (equivalently the KPZ equation) in a half-space with a repulsive and refecting (respectively) barrier and derives GSE Tracy–Widom limiting statistics. Both works are mathematically nonrigorous due to the ill-posed moment problem and certain unproved forms of the conjectured Bethe ansatz completeness of the half-space delta Bose gas. On the side of particle systems, half-space ASEP has first been studied in [41], but the resulting formulas were not amenable to asymptotic analysis. More recently, [9] derives exact Pfaffian formulas for the half-space stochastic six-vertex model, and proves GOE asymptotics for the current at the origin in half-space ASEP, for a certain boundary condition. The exact formulas involve the correlation kernel of the Pfaffian Schur process and the asymptotic analysis in [9]usessomeoftheideas developed here.

1.5. Main results. We now state the limit theorems that constitute the main results of this paper. They involve the GOE, GSE and GUE Tracy–Widom distributions, respectively, characterized by the distribution functions FGOE, FGSE and FGUE given in Section 2 and the standard Gaussian distribution function denoted by G(x).

THEOREM 1.3. The last passage time on the diagonal H(n, n) satisﬁes the following limit theorems, depending on the rate α of the weights on the diagonal: 1. For α>1/2, H(n, n) − 4n lim P

This extends the results of [5, 6] on the model with geometric weights. The first part of Theorem 1.3 is proved in Section 5, while the second and third parts are provedinSection6. Section 6.1 includes an explanation for why the transition occurs at α = 1/2, using a property of the Pfaffian Schur measure (Proposition 3.4). Far away from the diagonal, the limit theorem satisfied by H(n,m) coincides with that of the (unsymmetrized) full-quadrant model. √ κ HEOREM For any κ ∈ ( , ) and α> √ we have that T 1.4. 0 1 1+ κ , √ H(n,κn)− (1 + κ)2n lim P

This extends the results of [38] on the geometric model to the exponential case and to allow any boundary parameter. Theorem 1.4 is proved in Section 6.3.

REMARK 1.5. Our asymptotic analyses could be extended to show that the √ﬂuctuations√ of H(n,κn) for κ ∈ (0, 1) follow the BBP transition [3]whenα = + + O −1/3 2 κ/(1 √κ) (n√ ), and in particular are distributed according to (FGOE) when α = κ/(1 + κ). The BBP transition would also occur if one varies the rate of exponential weights in the ﬁrst rows of the lattice. Regarding H(n, n),itis not clear if the higher order phase transitions would coincide with the spiked GSE [12, 42].

Our results rely on an asymptotic analysis of the following formula for the joint distribution of passage times. It involves the Fredholm Pfaffian (see Section 2 for background on Fredholm Pfaffians) of a matrix-valued correlation kernel Kexp defined in Section 4.4 where the next proposition is proved.

PROPOSITION 1.6. For any h1,...,hk > 0 and integers 0 m2 > ···>mk such that ni >mi for all i, we have that k exp P H(n ,m )

In the one point case k = 1, and for n1 = m1, one could alternatively characterize the probability distribution of H(n, n) by taking the exponential limit of the formulas in [5, 6] using for instance the results from [1]. Note that we need only 3022 BAIK, BARRAQUAND, CORWIN AND SUIDAN the case k = 1 in order to prove Theorems 1.3 and 1.4, but the multipoint case is necessary for Theorems 1.7 and 1.8 stated below and proved in [2]. We can generalize the results of Theorems 1.3 and 1.4 by allowing the parameters κ and α to vary in regions of size n−1/3 around the critical values α = 1/2,κ = 1. In the limit, we obtain a new two-parametric family of probability distributions. More generally, we can compute the ﬁnite dimensional marginals of Airy-like processes in various ranges of α and κ. We state the results below and refer to [2] for detailed proofs. KPZ scaling predictions (together with Theorem 1.4) suggests to deﬁne H(n+ n2/3ξη,n− n2/3ξη)− 4n + n1/3ξ 2η2 H (η) = , n σn1/3 where η ≥ 0, σ = 24/3 and ξ = 22/3. Let us scale α as 1 + 2σ −1 n−1/3 α = , 2 where ∈ R is a free parameter. The limiting joint distribution of multiple points is characterized by a new crossover kernel Kcross that we introduce in Section 2.5.

≤ ··· ∈ R = 1+22/3 n−1/3 THEOREM 1.7. For 0 η1 < <ηk, , setting α 2 , we have that k cross lim P Hn(ηi)1/2 is ﬁxed, the joint distribution of passage-times is governed by the so-called symplectic-unitary transition [25].

THEOREM 1.8. For α>1/2 and 0 <η1 < ···<ηk, we have that SU lim P Hn(η1)

FIG.2. Phase diagram of the fluctuations of H(n,m)as n →∞when α and the ratio n/m varies. The gray area corresponds to a region of the parameter space where the fluctuations are on the scale n1/2 and Gaussian. The bounding curve (where fluctuations are expected to be Tracy–Widom GOE2 cf. Remark 1.5) asymptotes to zero as n/m goes to +∞. The crossover distribution Fcross(·; ,η) is defined in Definition 2.9 and describes the fluctuations in the vicinity of n/m = 1 and α = 1/2.

Theorem 1.8 can be seen as a →+∞degeneration of Theorem 1.7 (see Section 2.5). The proof is very similar with that of Theorem 1.7. We provide the details of the computations in [2]. An analogous result was found for the half-space PNG model without nucleations at the origin [38] although the statement of [38], Theorem 5.3, makes rigorous sense only when τ>0. The correlation kernel corresponding to the symplectic-unitary transition was studied ﬁrst in [25].8 This random matrix ensemble is the point process corresponding to the eigenvalues of a Hermitian complex matrix Xη for η ∈ (0, +∞) with density proportional to − − −η 2 exp Tr((Xη e X0) ) , 1−e−2η where X0 is a GSE matrix. Various limits of the symplectic-unitary and orthogonal- unitary transition kernels are considered in [25], Section 5. In particular, [25], Sec- tion 5.6, considers the limit as η → 0. While the GOE distribution is recovered in the orthogonal-unitary case, the explanations are missing in the symplectic-unitary case (the scaling argument at the end of [25], Section 5.6, is not the reason why one does not recover KGSE). We provide a rigorous proof in Section 6.5.

8 The integration domain in [25], (4.44), should be R<0 instead of R>0. The correct formula was found in [38], (5.39). 3024 BAIK, BARRAQUAND, CORWIN AND SUIDAN

REMARK 1.9. One can also study the limiting n-point distribution of √ H(n+ n2/3ξη,κn− n2/3ξη)− (1 + κ)2n − n1/3x2 η → , 24/3n1/3 for a ﬁxed κ ∈ (0, 1) and several values of x.Inthen →∞limit, one would obtain exp exp exp the extended Airy kernel for K12 and0forK11 and K22 but we do not pursue that direction.

Outline of the paper. In Section 2, we provide convenient Fredholm Pfaffian formulas for the Tracy–Widom distributions and their generalizations. In Sec- tion 3, we define Pfaffian Schur processes and construct dynamics preserving them, thus making a connection to half-space LPP (Proposition 3.10). In Section 4,we apply a general result of Borodin and Rains giving the correlation structure of Pfaf- fian Schur processes, in order to express the k-point distribution along space-like paths in half-space LPP with geometric (Proposition 4.2) and exponential (Propo- sition 1.6) weights. In Section 5, we discussed the issues related to the multiplicity of the limiting point process, and prove the first part of Theorem 1.3. In Section 6, we perform all other asymptotic analysis: we prove limit theorems toward the GUE Tracy–Widom distribution (Theorem 1.4), GOE Tracy–Widom distribution (second part of Theorem 1.3), and Gaussian distribution (third part of 1.3).

2. Fredholm Pfafﬁan formulas. Let us introduce a convenient notation that we use throughout the paper to specify integration contours in the complex plane.

ϕ DEFINITION 2.1. Let Ca be the union of two semi-inﬁnite rays departing a ∈ C with angles ϕ and −ϕ. We assume that the contour is oriented from a +∞e−iϕ to a +∞e+iϕ.

2.1. GUE Tracy–Widom distribution. For a kernel K : X × X → R,wedeﬁne + its Fredholm determinant det(I K)L2(X,μ) as given by the series expansion ∞ 1 k ⊗k det(I + K)L2 X = 1 + ··· det K(xi,xj ) = dμ (x1 ···xk), ( ,μ) k! X X i,j 1 k=1 whenever it converges. We will generally omit the measure μ in the notation and write simply L2(X) when the uniform or the Lebesgue measure is considered.

DEFINITION 2.2. The GUE Tracy–Widom distribution, denoted LGUE is a probability distribution on R such that if X ∼ LGUE, P ≤ = = − (X x) FGUE(x) det(I KAi)L2(x,+∞), where KAi is the Airy kernel, 3 ez /3−zu 1 (1) KAi(u, v) = dw dz . C2π/3 Cπ/3 w3/3−wv − −1 1 e z w PFAFFIAN SCHUR PROCESSES AND LPP IN A HALF-QUADRANT 3025

Throughout the paper, all integrals over a contour of the complex plane will take a := 1 factor 1/(2iπ), and thus we use the notation dz 2iπ dz.

2.2. Fredholm Pfaffian. In order to define the GOE and GSE distribution in a form which is convenient for later purposes, we introduce the concept of Fredholm Pfaffian. We refer to Section 4.1 explaining how Fredholm Pfaffians naturally arise in the study of Pfaffian point processes.

DEFINITION 2.3 ([37] Section 8). For a 2 × 2-matrix valued skew-symmetric kernel, K (x, y) K (x, y) K(x,y) = 11 12 ,x,y∈ X K21(x, y) K22(x, y) × k [such kernel is called skew-symmetric if the 2k 2k matrix (K(xi,xj ))i,j=1 is skew-symmetric] we define its Fredholm Pfaffian by the series expansion ∞ 1 k ⊗k (2) Pf(J + K)L2 X = 1 + ··· Pf K(xi,xj ) = dμ (x1 ···xk), ( ,μ) k! X X i,j 1 k=1 provided the series converges, and we recall that for an skew-symmetric 2k × 2k matrix A, its Pfaffian is defined by 1 (3) Pf(A) = sign(σ )a a ···a − . 2kk! σ(1)σ(2) σ(3)σ(4) σ(2k 1)σ(2k) σ∈S2k The kernel J is defined by 01 J(x,y) = 1 = . x y −10

REMARK 2.4. We must consider matrix kernels as made up of k2 blocks, each of which has size 2 × 2. Considering 22 blocks of size k × k instead would change the value of its Pfafﬁan by a factor (−1)k(k−1)/2.

In Sections 4, 5 and 6, we will need to control the convergence of Fredholm Pfafﬁan series expansions. This can be done using Hadamard’s bound. We recall that for a matrix M of size k,ifthe(i, j) entry of M is bounded by aibj for all i, j ∈{1,...,k} then k k/2 det[M] ≤ k aibi. i=1 √ Using this inequality and the fact that Pf[A]= det[A] for a skew-symmetric matrix A, we have the following. 3026 BAIK, BARRAQUAND, CORWIN AND SUIDAN

LEMMA 2.5. Let K(x,y) a 2 × 2 matrix valued skew symmetric kernel. As- sume that there exist constants C>0 and constants a>b≥ 0 such that −ax−ay −ax+by K11(x, y)

Then, for all k ∈ Z>0, k k k/2 k −(a−b)xi Pf K(xi,xj ) i,j=1 <(2k) C e . i=1

2.3. GOE Tracy–Widom distribution. The GOE Tracy–Widom distribution, denoted LGOE, is a continuous probability distribution on R. The following is a convenient formula for its cumulative distribution function.

∼ L := P ≤ = − GOE LEMMA 2.6. For X GOE, FGOE(x) (X x) Pf(J K )L2(x,∞), where KGOE is the 2 × 2 matrix valued kernel deﬁned by z − w 3 + 3 − − KGOE(x, y) = dz dw ez /3 w /3 xz yw, 11 Cπ/3 Cπ/3 z + w 1 1 w − z 3 + 3 − − KGOE(x, y) =−KGOE(x, y) = dz dw ez /3 w /3 xz yw, 12 21 Cπ/3 Cπ/3 + 1 −1/2 2w(z w) z − w 3 + 3 − − KGOE(x, y) = dz dw ez /3 w /3 xz yw 22 Cπ/3 Cπ/3 + 1 1 4zw(z w) 3 − dz 3 − dz sgn(x − y) + ez /3 zx − ez /3 zy − . Cπ/3 Cπ/3 1 4z 1 4z 4 Throughout the paper, we adopt the convention that

sgn(x − y) = 1x>y − 1x

PROOF. According to [40] (see also equivalent formulas [23], (2.9) and 1 (6.17)), the GOE distribution function can be defined by FGOE = Pf(J − K ), where K1 is defined by the matrix kernel ∞ ∞ 1 = + + − + + K11(x, y) dλAi(x λ)Ai (y λ) dλAi(y λ)Ai (x λ), 0 0 ∞ 1 =− 1 = + + K12(x, y) K21(y, x) Ai(x λ)Ai(y λ) 0 1 ∞ + Ai(x) dλAi(y − λ), 2 0 ∞ ∞ 1 = 1 + + K22(x, y) dλ dμAi(y μ)Ai(x λ) 4 0 λ PFAFFIAN SCHUR PROCESSES AND LPP IN A HALF-QUADRANT 3027 1 ∞ ∞ − dλ dμAi(x + μ)Ai(y + λ) 4 0 λ 1 +∞ 1 +∞ sgn(x − y) − Ai(x + λ) dλ + Ai(y + λ) dλ − . 4 0 4 0 4 Using the contour integral representation of the Airy function 3 − (4) Ai(x) = ez /3 zx dz for any ϕ ∈ (π/6,π/2] and a ∈ R , ϕ >0 Ca Cπ/3 + + with integration on 1 for Ai(x λ) and Ai (x λ), we readily see that ∞ 3 + 3 − − − − K1 (x, y) = dλ dz dw(z − w)ez /3 w /3 xz yw λz λw 11 Cπ/3 Cπ/3 0 1 1 ∞ 3 + 3 − − − − (5) = dz dw(z − w)ez /3 w /3 xz yw dλe λz λw Cπ/3 Cπ/3 1 1 0 = GOE K11 (x, y). The exchange of integrations in (5) is justified because z + w has positive real part GOE Cπ/3 along the contours. Turning to K12 , we first deform the contour −1/2 for the Cπ/3 variable w to 1 . When doing this, we encounter a pole at zero. Thus, taking into account the residue at zero, we find that 1 z − w 3 + 3 − − KGOE(x, y) = Ai(x) + dz dw ez /3 w /3 xz yw. 12 Cπ/3 Cπ/3 + 2 1 1 2w(z w) Cπ/3 Using again (4) with the contour 1 ,wefindthat ∞ ∞ GOE = 1 + + + − 1 + K12 (x, y) Ai(x) Ai(x λ)Ai(y λ) dλ Ai(x) dλAi(y λ) 2 0 2 0 = 1 K12(x, y), +∞ = 1 where in the last equality we have used that 0 Ai(λ) dλ 1. For K22 we use Cπ/3 + + similarly (4) with integration on 1 for Ai(x λ) and Ai(x μ) and get that 1 = GOE K22(x, y) K22 (x, y).

2.4. GSE Tracy–Widom distribution. The GSE Tracy–Widom distribution, denoted LGSE, is a continuous probability distribution on R.

∼ LGSE := P ≤ = − GSE LEMMA 2.7. For X , FGSE(x) (X x) Pf(J K )L2(x,∞), where KGSE is a 2 × 2-matrix valued kernel defined by z − w 3 + 3 − − KGSE(x, y) = dz dw ez /3 w /3 xz yw, 11 Cπ/3 Cπ/3 + 1 1 4zw(z w) 3028 BAIK, BARRAQUAND, CORWIN AND SUIDAN z − w 3 + 3 − − KGSE(x, y) =−KGSE(x, y) = dz dw ez /3 w /3 xz yw, 12 21 Cπ/3 Cπ/3 4z(z + w) 1 1 z − w 3 + 3 − − KGSE(x, y) = dz dw ez /3 w /3 xz yw. 22 Cπ/3 Cπ/3 + 1 1 4(z w) = − 4 PROOF. FGSE(x) is defined as FGSE(x) det(I K )L2(x,∞) in [40], Sec- tion III, where K4 is the 2 × 2 matrix valued kernel defined by ∞ 4 = 4 = 1 − 1 K11(x, y) K22(y, x) KAi(x, y) Ai(x) Ai(λ) dλ, 2 4 y −1 1 K4 (x, y) = ∂ K (x, y) − Ai(x)Ai(y), 12 2 y Ai 4 − ∞ ∞ ∞ 4 = 1 + 1 K21(x, y) KAi(λ, y) dλ Ai(λ) dλ Ai(μ) dμ. 2 x 4 x y In order to have a Pfaffian formula, we need a skew-symmetric kernel. We compute the kernel KGSE := JK4 using 01ab cd = , −10 cd −a −b and find that the operator KGSE is given by the matrix kernel GSE = 4 K11 (x, y) K21(x, y), GSE = 4 K12 (x, y) K22(x, y), GSE =− 4 =− 4 =− GSE K21 (x, y) K11(x, y) K22(y, x) K12 (y, x), GSE =− 4 K22 (x, y) K12(x, y). Using the contour integral representations for the Airy function (4) with integra- Cπ/3 tions on 1 and the definition of the Airy kernel (1), we find that the entries GSE GSE Kij (x, y) match those given in the statement of Lemma 2.7.NowK is skew- symmetric. Using that for a skew-symmetric kernel A,Pf(J + A)2 = det(I − JA) as soon as Fredholm expansions are convergent ([37], Lemma 8.1) (see also [36], Proposition B.4, for a proof), we have that (using J 2 = I ) = − 4 = − GSE FGSE(x) det I K L2(x,∞) Pf J K L2(x,∞).

REMARK 2.8. There exists an alternative scalar kernel which yields the GSE distribution function [28]: = − GSE = − GLD (6) FGSE(x) Pf J K L2(x,∞) det I K L2(x,∞), PFAFFIAN SCHUR PROCESSES AND LPP IN A HALF-QUADRANT 3029

where ∞ GLD 1 K (x, y) = KAi(x, y) − Ai(x) dzAi(y + z). 2 0 The identity (6) can be shown by factoring KGSE and using the det(I + AB) = det(I + BA) trick.

2.5. Crossover kernels. The crossover kernel in Theorem 1.7 concerns the limiting ﬂuctuations of multiple points, hence the kernel is indexed by elements of {1,...,k}×R. We introduce a matrix kernel Kcross, depending on parameters ∈ R and 0 ≤ η1 < ···<ηk, which decomposes as Kcross(i, x; j,y) = I cross(i, x; j,y) + Rcross(i, x; j,y). We have z + η − w − η I cross(i, x; j,y) = dz dw i j 11 Cπ/3 Cπ/3 + + + 1 1 z w ηi ηj z + + η w + + η 3 + 3 − − × i j ez /3 w /3 xz yw, z + ηi w + ηj z + η − w + η I cross(i, x; j,y) = dz dw i j 12 Cπ/3 Cπ/3 + + + − az aw 2(z ηi)(z ηi w ηj ) z + + η 3 + 3 − − × i ez /3 w /3 xz yw, −w + + ηj cross ; =− cross ; I21 (i, x j,y) I12 (j, y i, x), z − η − w + η I cross(i, x; j,y) = dz dw i j 22 Cπ/3 Cπ/3 4(z − η + w − η ) bz bw i j 3 3 ez /3+w /3−xz−yw × . (z − − ηi)(w − − ηj ) cross − + − + The contours in I12 are chosen so az > ηi , az aw >ηj ηi and aw < ηj . cross + + The contours in I22 are chosen so bz >ηi , bz >ηi and bw >ηj , bw >ηj cross ; = cross ; = ≥ .WehaveR11 (i, x j,y) 0, and R12 (i, x j,y) 0wheni j.When i

4 2 2 −(ηi −ηj ) +6(x+y)(ηi −ηj ) +3(x−y) − exp( − ) cross ; = 12(ηi ηj ) R12 (i, x j,y) , 4π(ηj − ηi) that we may also write +∞ cross −λ(ηi −ηj ) R (i, x; j,y) =− dλe Ai(xi + λ)Ai(xj + λ). 12 −∞ 3030 BAIK, BARRAQUAND, CORWIN AND SUIDAN

cross − − The kernel R22 is antisymmetric, and when xi ηi >xj ηj we have + 3 + + 3 − + − + −1 e(z ηi ) /3 ( ηj ) /3 x(z ηi ) y( ηj ) Rcross(i, x; j,y) = dz 22 Cπ/3 + 4 az z + 3 + + 3 − + − + 1 e(z ηj ) /3 ( ηi ) /3 y(z ηj ) x( ηi ) + dz Cπ/3 + 4 az z + 3 + − + 3 − + − − + 1 ze(z ηi ) /3 ( z ηj ) /3 x(z ηi ) y( z ηj ) − dz , 2 Cπ/3 ( + z)( − z) bz where the contours are chosen so that az < − and −| |

DEFINITION 2.9. We deﬁne the probability distribution Lcross by the distribution function ; = − cross ·; · Fcross(x ,η) Pf J K (1, 1, ) L2(x,∞).

This family of distribution functions realizes a two-dimensional crossover between GSE, GUE and GOE distributions, in the sense that Fcross(x; 0, 0) = FGOE(x) and

Fcross(x; ,η) ⎧ ⎪F (x) when →+∞and η = 0, ⎪ GSE ⎨ 2 →−∞ →+∞ =− −→ FGOE(x) when ,η with /η 1, ⎪ ⎪FGUE(x) when η →+∞and + η →+∞, ⎩⎪ 0when →−∞and + η →−∞. Moreover, when →−∞and + η →−∞, y 2 1 − 1 s2 Fcross ( + η) + 2| + η|y; ,η −→ √ e 2 ds. −∞ 2π

In addition, for a ∈ R, Fcross(x; ,η) −→ F1(x; a) when →−∞,η→+∞ with + η = a,whereF1(x; a) is the distribution in [3], Definition 1.3, that 2 interpolates FGUE(x) and (FGOE(x)) .Whenη = 0, Fcross(x; ,0) = F(x; ) where the crossover distribution F(x; w) is defined in [6], Definition 4. The crossover kernel arising in Theorem 1.8 is a matrix kernel KSU, depending on parameters ∈ R and 0 <η1 < ···<ηk, of the form KSU(i, x; j,y) = I SU(i, x; j,y) + RSU(i, x; j,y). We have 3 3 (z + η − w − η )ez /3+w /3−xz−yw I SU(i, x; j,y) = dz dw i j , 11 Cπ/3 Cπ/3 + + + + + 1 1 4(z ηi)(w ηj )(z w ηi ηj ) PFAFFIAN SCHUR PROCESSES AND LPP IN A HALF-QUADRANT 3031

3 3 (z + η − w + η )ez /3+w /3−xz−yw I SU(i, x; j,y) = dz dw i j , 12 Cπ/3 Cπ/3 + + + − az aw 2(z ηi)(z w ηi ηj ) SU ; =− SU ; I21 (i, x j,y) I12 (j, xj i, yi), z − η − w + η 3 + 3 − − I SU(i, x; j,y) = dz dw i j ez /3 w /3 xz yw. 22 Cπ/3 Cπ/3 z − η + w − η bz bw i j SU − + − The contours in I12 are chosen so az > ηi , az aw >ηj ηi . The contours in SU I22 are chosen so that bz >ηi and bw >ηj . SU ; = SU ; = ≥ We have R11 (i, x j,y) 0, and R12 (i, x j,y) 0wheni j.Wheniy ηj we have 1 + 3 + − + 3 − + − − + RSU(i, x; j,y) =− dzze(z ηi ) /3 ( z ηj ) /3 x(z ηi ) y( z ηj ), 22 Cπ/3 2 0 where the contours are chosen so that az > − and bz is between − and . Modulo a conjugation of the kernel, KSU is the limit of Kcross when goes to +∞,thatis, ⎛ ⎞ 1 cross cross SU ⎝ K K ⎠ K = lim 4 2 11 12 . →∞ cross 2 cross K21 4 K22

3. Pfaffian Schur process. The key tool in our analysis of last passage percolation on a half-space is the Pfaffian Schur process. In this section, we first introduce the Pfaffian Schur process as in [21]. This is a Pfaffian analogue of the determinantal Schur process introduced in [35]. We refer to [19], Section 6, and references therein for background on the Schur process. Then, in order to connect it to last passage percolation, we introduce an equivalent presentation of the Pfaf- fian Schur processes indexed by lattice paths in the half-quadrant, and we study dynamics preserving the Pfaffian Schur process.

3.1. Deﬁnition of the Pfafﬁan Schur process.

3.1.1. Partitions and Schur positive specializations. Pfafﬁan Schur processes are measures on sequences of integer partitions. A partition is a nonincreasing sequence of nonnegative integers λ = (λ1 ≥ λ2 ≥···≥λk ≥ 0), with ﬁnitely many nonzero components. Given two partitions λ,μ, we write μ ⊂ λ if λi ≥ μi for all i ≥ 1. We write μ ≺ λ and say that μ interlaces λ if for all i ≥ 1, λi ≥ μi ≥ λi−1. 3032 BAIK, BARRAQUAND, CORWIN AND SUIDAN

For a given partition λ, its dual, denoted λ , is the partition corresponding to the Young diagram which is symmetric to the Young diagram of λ with respect to the = { : ≥ } ﬁrst diagonal. In other words, λi j λj i . We say that a partition λ is even if all its component are even integers. Thus, for a partition λ, we say that its dual

λ is even when we have λ1 = λ2,λ3 = λ4,etc.Wedenoteby|λ| the number of boxes in the diagram corresponding to λ. The probabilities in the Pfaffian Schur process—and the usual Schur process as well—are expressed in terms of skew Schur symmetric functions sλ/μ indexed by pairs of partitions μ,λ. We use the convention that sλ/μ = 0ifμ ⊂ λ. We refer to [19], Section 2, for a definition of (skew) Schur functions and background on symmetric functions. We record later in Section 3.1.2 the few properties of Schur functions that we will use. A specialization ρ of the algebra Sym of symmetric functions is an algebra morphism from Sym to C. For example, the evaluation of a symmetric function into one fixed variable α ∈ C defines such a morphism. We denote f(ρ)the application of the specialization ρ to f ∈ Sym. A specialization can be defined by its values on a basis of Sym and we generally choose the power sum symmetric functions = n pn(x1,x2,...) xi . For two specializations ρ1 and ρ2, their union, denoted (ρ1,ρ2),isdefinedby

pn(ρ1,ρ2) = pn(ρ1) + pn(ρ2) for all n ≥ 1.

More generally, the specialization (ρ1,...,ρn) is the union of specializations ρ1,...,ρn.Whenρ1 and ρ2 are specializations corresponding to the evaluation into sets of variables, then (ρ1,ρ2) corresponds to the evaluation into the union of both sets of variables, hence the term union for this operation on specializations. Schur nonnegative specializations of Sym are specializations taking values in R≥0 when applied to skew Schur functions sλ/μ for any partitions λ and μ. Thoma’s theorem (see [30] and references therein) provides a classiﬁcation of such = = specializations. Let α (αi)i≥1, β (βi)i≥1 and γ be nonnegative numbers such that (αi + βi)<∞. Any Schur nonnegative specialization ρ is determined by parameters (α,β,γ)through the formal series identity

1 + β z h (ρ)zn = exp(γ z) i =: H(z; ρ), n 1 − α z n≥0 i≥1 i where the hn are complete homogeneous symmetric functions. The specialization (0, 0,γ) is called (pure-)Plancherel and will be denoted by Plancherel(γ ). It can be obtained as a limit of specializations (α, 0, 0) where α is the sequence of length M, (γ /M, . . . , γ /M),whenM is sent to inﬁnity. From now on, we assume that all specializations are always Schur nonnegative. PFAFFIAN SCHUR PROCESSES AND LPP IN A HALF-QUADRANT 3033

3.1.2. Useful identities. We collect some identities from [31]thatwereused in [21] to deﬁne the Pfafﬁan Schur process. The sums below [as in (7)] are always taken over all partitions, unless otherwise restricted. Starting with one already useful for the determinantal Schur process, we have the skew-Cauchy identity (7) sν/λ(ρ)sν/μ ρ = H ρ; ρ sλ/τ ρ sμ/τ (ρ), ν τ

where, if ρ and ρ are the specializations into sets of variables {xi}, {yi}, 1 H ρ; ρ = . − i,j 1 xiyj We also use the branching rule for Schur functions (8) sλ/ν(ρ)sν/μ ρ = sλ/μ ρ,ρ . ν The following identity is crucial to go from the determinantal Schur processes to its Pfafﬁan analog. It is a variant of the skew-Littlewood identity, ◦ (9) sν/λ(ρ) = H (ρ) sλ/κ(ρ), ν even κ even where if ρ is the specialization into a set of variables {xi}, ◦ 1 H (ρ) = . − i

3.1.3. Deﬁnition of the Pfafﬁan Schur process.

DEFINITION 3.1 ([21]). The Pfafﬁan Schur process parametrized by two se- + + + − − quences of Schur nonnegative specializations ρ0 ,ρ1 ,...,ρn−1 and ρ1 ,...,ρn , [ +; +; ; + | −; ; −] denoted PSP ρ0 ρ1 ... ρn−1 ρ1 ... ρn , is a probability measure on sequences of integer partitions − ∅ ⊂ λ(1) ⊃ μ(1) ⊂ λ(2) ⊃ μ(2) ···μ(n 1) ⊂ λ(n) ⊃ ∅. Under this measure, the probability of the sequence λ¯ = (λ(1),...,λ(n)), μ¯ = (μ(1),...,μ(n)) is given by ¯ + + + − − V(λ, μ)¯ PSP ρ ; ρ ; ...; ρ |ρ ; ...; ρ (λ,¯ μ)¯ = , 0 1 n−1 1 n Z◦(ρ) 3034 BAIK, BARRAQUAND, CORWIN AND SUIDAN

(1) (2) (n) λ − + λ + λ + ρ ρ ρ − ρ0 1 1 n−1 ρn ··· ∅ μ(1) μ(n−1) ∅

[ +; +; ; + | −; ; FIG.3. Diagram corresponding to The Pfafﬁan Schur process PSP ρ0 ρ1 ... ρn−1 ρ1 ... −] ρn . The ascending and descending links represent inclusions of partitions. where Z◦(ρ) is a renormalization constant depending on the choice of specializations and the weight V(λ,¯ μ)¯ is given by V ¯ ¯ = + − + ×··· (λ, μ) τλ(1) ρ0 sλ(1)/μ(1) ρ1 sλ(2)/μ(1) ρ1 (10) × + − sλ(n)/μ(n−1) ρn−1 sλ(n) ρn , where τλ = sλ/κ. κ even

We may encode the choice of specializations by the diagram shown in Figure 3. The normalization constant Z◦(ρ) for the weights V(λ,¯ μ)¯ is ◦ Z (ρ) := V(λ,¯ μ),¯ ∅⊂λ(1)⊃μ(1)⊂λ(2)⊃μ(2)···μ(n−1)⊂λ(n)⊃∅ and can be computed [21]tobe ◦ = ◦ − − +; − Z (ρ) H ρ1 ,...,ρn H ρi ρj . 0≤i

DEFINITION 3.2. The Pfafﬁan Schur measure PSM[ρ|ρ ] is a probability measure on a single partition λ such that

1 PSM ρ|ρ (λ) = τ (ρ)s ρ , H ◦(ρ )H (ρ; ρ ) λ λ

(k) [ +; +; LEMMA 3.3. The law of λ under the Pfafﬁan Schur process PSP ρ0 ρ1 ; + | −; ; −] [ + + | − ... ρn−1 ρ1 ... ρn is the Pfafﬁan Schur measure PSM ρ0 ,...,ρk−1 ρk ,..., −] ρn .

PROOF. Summing over all partitions except λ(k) in (10) and using identities (7), (8)and(9) yields the result.

The following property of the Pfafﬁan Schur measure will be useful to interpret the results in Section 6.1. PFAFFIAN SCHUR PROCESSES AND LPP IN A HALF-QUADRANT 3035

PROPOSITION 3.4 (Corollary 7.6 [5]). Let c be a positive real and ρ a Schur nonnegative specialization. If μ is distributed according to PSM[c|ρ] and λ is (d) distributed according to PSM[0|c,ρ], then we have the equality in law λ1 = μ1.

PROOF. By deﬁnition, we have that τ (0)s (c, ρ) P(λ ≤ x) = λ λ , 1 H ◦(c, ρ) λ:λ1≤x (11) τ (c)s (ρ) P(μ ≤ x) = μ μ . 1 H ◦(ρ)H (c; ρ) μ:μ1≤x ◦ ◦ The normalization constants H (c, ρ) and H (ρ)H (c; ρ) are equal. Since τλ(0) = 1λ even,wehavethat 1 LHS (11) = s (c, ρ) H ◦(ρ)H (c; ρ) λ λ:λ1≤x λ even 1 = s (ρ)s (c) (Branching rule) H ◦(ρ)H (c; ρ) μ λ/μ λ:λ1≤x μ λ even 1 − + − +··· = s (ρ)cμ1 μ2 μ3 μ4 H ◦(ρ)H (c; ρ) μ μ:μ1≤x 1 = s (ρ)τ (c) H ◦(ρ)H (c; ρ) μ μ μ:μ1≤x = RHS (11).

Note that in the third equality, we use that if λ is even and μ ≺ λ, then we have for all i ≥ 1, λ2i−1 = μ2i−1 = λ2i . Thus, the fact that c is a single variable specialization is necessary. The fourth equality comes from the fact that for a single variable specialization c, μ1−μ2+μ3−μ4+··· (12) τμ(c) = sλ/κ(c) = c . κ even Indeed, there is only one partition κ which gives a nonzero contribution in the sum in (12).

REMARK 3.5. The proof of Proposition 3.4 actually shows that (λ1,λ3, λ5,...) has the same law as (μ1,μ3,μ5,...), but we will use only the ﬁrst coordinates in applications. 3036 BAIK, BARRAQUAND, CORWIN AND SUIDAN

FIG.4. A zig-zag path indexing the Pfafﬁan Schur process.

3.2. Pfaffian Schur processes indexed by zig-zag paths. In Definition 3.1, Pfaffian Schur processes were determined by two sequences of specializations + + − − ρ◦ ,...,ρn−1 and ρ1 ,...,ρn . We describe here an equivalent way of represent- ing Pfaffian Schur processes, indexing them by certain zig-zag paths in the first quadrant of Z2 (as depicted in Figure 4) where each edge of the path is labelled by some Schur nonnegative specialization. This representation is convenient for defining dynamics on Pfaffian Schur processes. More precisely, we consider oriented paths starting on the horizontal axis at (+∞, 0), proceeding by unit steps horizontally to the left or vertically to the top, hitting the diagonal at√ some point (n, n) for some n ∈ Z>0 and then taking one final edge of length n 2 to the origin (0, 0) (see Figure 4). Let us denote the set of all such paths. For γ ∈ , let us denote E(γ) its set of edges and V(γ)its set of vertices. We denote E↑(γ ) the set of vertical edges and E←(γ ) the set of horizontal edges. Each edge e is labelled by a Schur nonnegative specialization ρe. We label by a specialization ρ◦ the edge joining the point (n, n) to (0, 0). Each vertex v ∈ γ indexes a partition λv. For γ ∈ , the Pfaffian Schur process indexed by γ and the specializations on v E(γ) is a probability measure on the sequence of partitions λ := (λ )v∈V(γ) where the weight of λ is given by V λ = W ( ) τλ(n,n) (ρ◦) (e), e∈E↑(γ )∪E←(γ ) ← ↑ where for an edge e1 ∈ E (π) and for an edge e2 ∈ E (π), v W = ← = u v W = ↑ = v u e1 v u sλ /λ (ρe1 ) and e2 sλ /λ (ρe2 ). u We adopt the convention that for all vertices v on the horizontal axis, λv is the empty partition, which in particular implies that specializations are empty on the PFAFFIAN SCHUR PROCESSES AND LPP IN A HALF-QUADRANT 3037

horizontal axis. We also recall that sλ/μ ≡ 0ifμ ⊂ λ. Using the above formalism, the normalisation constant associated to the weights V(λ) is given by ◦ ◦ Z (γ ) = H (ρe)H (ρ◦; ρe) H(ρe; ρe ), e∈E↑(γ ) e∈E↑(γ ),e ∈E←(γ ) e>e where e>e means that the edge e occurs after the edge e in the oriented path γ .

REMARK 3.6. The above description is equivalent to the one given in Defini- tion 3.1. As we have defined in Section 3.1, the Pfaffian Schur process is a measure on sequences − ∅ ⊂ λ(1) ⊃ μ(1) ⊂ λ(2) ⊃ μ(2) ···μ(n 1) ⊂ λ(n) ⊃ ∅. However, by using empty specializations, one can force that λ(i) = μ(i) or μ(i) = λ(i+1). Hence, we can consider sequences where the order of inclusions is any word on the alphabet {⊂, ⊃}. By matching the inclusions ⊃ with vertical edges in the zig-zag path formulation and the inclusions ⊂ with horizontal edges, we see that both formulations are equivalent (zig-zag paths are in bijection with finite words over a two-letter alphabet); see, for instance, Figure 6 for a particular example. This zig-zag construction also applies to the usual Schur process [35]. In that case, the processes are indexed by paths from a point on the horizontal axis to a point on the vertical axis.

3.3. Dynamics on Pfaffian Schur processes. Here, we give a sequential construction of Pfaffian Schur processes using the formalism of Section 3.2 by defining Markov chains preserving the Pfaffian Schur structure. More precisely, we will define Markov dynamics such that the pushforward of the Pfaffian Schur process indexed by a path γ ∈ and a set of specializations on E(γ) is the Pfaffian Shur process indexed by a path γ and the same set of specializations, where γ is obtained from γ by adding one box to the shape it defines, or half a box when it grows along the diagonal. These dynamics are a special case of those developed in [8] with Alexei Borodin in the Macdonald case [15]. They are an adaptation to the Pfaffian setting of Markov dynamics preserving the (determinantal) Schur process introduced in [17], Section 2, and studied in a more general setting in [13](seethe review [19], Section 6.4). Let us explain precisely how to obtain a path γ ∈ as the outcome of a sequence of elementary moves (see Figure 5). (i) We start with the path with only one vertex (0, 0). (ii) We may first make the path grow along the diagonal and get (1, 0) → (1, 1) → (0, 0). (iii) We can always freely move to the right the starting point of the path. For instance, assume that we get the new path (2, 0) → (1, 0) → (1, 1) → (0, 0). (iv) Then we may make the path grow by one box and get (2, 0) → (2, 1) → (1, 1) → (0, 0). (v) We will continue iteratively adding boxes to the path or growing along the diagonal, until we arrive at γ . 3038 BAIK, BARRAQUAND, CORWIN AND SUIDAN

FIG.5. Illustration of the ﬁrst steps according to which the path grows.

In a parallel way, we construct a sequence of Pfaffian Schur processes for each path described above. (i) We start with an empty Pfaffian Schur process λ(0,0) = ∅. (ii) The application of Markov dynamics—that we shall define momentarily— corresponding to the growth of the path along the diagonal will define a Pfaffian Schur process on ∅ = λ(1,0) ⊂ λ(1,1) ⊃ λ(0,0) = ∅. (iii) By convention, the partitions indexed by vertices on the horizontal axis are empty under the Pfaffian Schur process. Hence, sliding the starting point of the path to the right has no effect in terms of Pfaffian Schur process. (iv) Then the application of Markov dynamics corresponding to the growth of the path by one box at the corner (1, 0) will define a Pfaffian Schur process ∅ = λ(2,0) ⊂ λ(2,1) ⊃ λ(1,1) ⊃ λ(0,0) = ∅. (v) We continue iteratively by applying Markov operators that update one partition in the Pfaffian Schur process. We now have to define the above-mentioned Markov dynamics, and explain how the specializations are chosen. We have seen that there are two distinct cases corresponding to the growth of the path by one box or the growth by a half-box along the diagonal. We define a transition operator U corresponding to the growth of the path ρ1,ρ2 by one box, as a probability distribution U (π|ν,μ,κ) on partitions π,given ρ1,ρ2 partitions ν,μ,κ with μ ⊂ ν,κ. In terms of growing paths, this operator corresponds to adding one box in a corner formed by partitions μ ⊂ ν,κ, and it will update the partition μ to a partition π containing ν and κ, in such a way that the corner formed by partition κ,π,ν is the marginal of a new Pfaffian Schur process. Pictorially, the action of the transition operator U can be represented by the ρ1,ρ2 following diagram:

U updates the Pfafﬁan Schur process formed by partitions κ ⊃ μ ⊂ ν on the ρ1,ρ2 left to the the Pfafﬁan Schur process formed by partitions κ ⊂ π ⊃ ν on the right. After the update, one can forget the information supported by the gray arrows on PFAFFIAN SCHUR PROCESSES AND LPP IN A HALF-QUADRANT 3039

the right. The specializations in the new Pfafﬁan Schur process are carried from the previous step by the transition operator as shown above.

LEMMA 3.7. Assume that U satisfies ρ1,ρ2 s (ρ )s (ρ ) (13) s (ρ )s (ρ )U (π|ν,μ,κ) = π/κ 1 π/ν 2 . κ/μ 2 ν/μ 1 ρ1,ρ2 ; μ H(ρ2 ρ1) Then U preserves the Pfaffian Schur process measure in the following sense. ρ1,ρ2 Let γ ∈ contain the corner defined by the vertices ρ ρ (i + 1,j)−→2 (i, j) −→1 (i, j + 1),

where ρ1 and ρ2 are the specializations indexing edges of the corner. Let γ ∈ contain the same set of vertex as γ except that the vertex (i, j) is replaced by the vertex (i + 1,j + 1), and assume that the specializations are now chosen as ρ ρ (i + 1,j)−→1 (i + 1,j + 1) −→2 (i, j + 1).

Then U (λ(i+1,j+1)|λ(i,j+1),λ(i,j),λ(i+1,j)) maps the Pfafﬁan Schur process ρ1,ρ2 indexed by γ and the set of specializations on E(γ ) to the Pfafﬁan Schur process indexed by γ and the same set of specializations.

PROOF. The relation (13) implies that applying U to the Pfafﬁan Schur ρ1,ρ2 process indexed by γ and averaging over λ(i,j) gives a weight proportional to

sλ(i+1,j+1)/λ(i+1,j) (ρ1)sλ(i+1,j+1)/λ(i,j+1) (ρ2), as in the Pfafﬁan Schur process indexed by γ . The normalization H(ρ2; ρ1) ac- ◦ ◦ counts for the fact that Z (γ ) = H(ρ2; ρ1)Z (γ ).

We choose a particular solution to (13)givenby

s (ρ )s (ρ ) (14) U (π|ν,μ,κ) = U (π|ν,κ) = π/ν 2 π/κ 1 . ρ1,ρ2 ρ1,ρ2 λ sλ/ν(ρ2)sλ/κ(ρ1) The Cauchy identity (7) ensures that this choice satisfies (13). There exists other solutions to (13)(see[20, 32]) where U (π|ν,μ,κ) depends on μ. We also define a diagonal transition operator U ∠ corresponding to the ρ◦,ρ1 growth of the path by a half box along the diagonal as a probability distribution U ∠ (π|κ,μ) on partitions π given partitions μ ⊂ κ. It will update the partition ρ◦,ρ1 μ ⊂ κ to a partition π ⊃ κ such that π is a marginal of a new Pfaffian Schur process. Pictorially, 3040 BAIK, BARRAQUAND, CORWIN AND SUIDAN

Again, the specializations in the new Pfafﬁan Schur process are carried from the previous step by the transition operator.

∠ LEMMA 3.8. Assume that U satisfies ρ◦,ρ1 ∠ sπ/κ(ρ1)τπ (ρ◦) (15) s (ρ )τ (ρ◦)U (π|κ,μ) = . κ/μ 1 μ ρ◦,ρ1 ◦ ; μ H (ρ1)H (ρ1 ρ◦) Let γ ∈ contain the vertices (n, n) and (n + 1,n), with the edge (n + 1,n)→ (n, n) labelled by a specialization ρ1 and the diagonal edge labelled by ρ◦ as usual. Let γ ∈ contain the vertices (n + 1,n+ 1) and (n + 1,n), with the edge (n + 1,n)→ (n + 1,n+ 1) labelled by the specialization ρ1 and the diagonal ∠ + + + edge labelled by ρ◦. Then U (λ(n 1,n 1)|λ(n 1,n),λ(n,n)) maps the Pfaffian Schur process indexed by γ and the set of specializations on E(γ ) to the Pfaffian Schur process indexed by γ and the same set of specializations.

∠ PROOF.(15) implies that applying U to the Pfafﬁan Schur process in- ρ◦,ρ1 dexed by γ and averaging over λ(n,n) yields a weight proportional to

sλ(n+1,n+1)/λ(n+1,n) (ρ1)τλ(n+1,n+1) (ρ◦) ◦ as in the Pfafﬁan Schur process indexed by γ . The normalization H (ρ1)H (ρ1; ◦ ◦ ◦ ρ◦) accounts for the fact that Z (γ ) = H (ρ1)H (ρ1; ρ◦)Z (γ ).

We choose a particular solution to (15)givenby

∠ ∠ 1 τ (ρ◦)s (ρ ) (16) U (π|κ,μ) = U (π|κ)= π π/κ 1 . ρ◦,ρ1 ρ◦,ρ1 ◦ H (ρ1)H (ρ1; ρ◦) τκ (ρ◦,ρ1) ◦ ◦ Let us check that (15) is satisﬁed. Denoting Z = H (ρ1)H (ρ1; ρ◦), ∠ s (ρ )τ (ρ◦)U (π|κ,μ) κ/μ 1 μ ρ◦,ρ1 μ = 1 τπ (ρ◦)sπ/κ(ρ1) ◦ sκ/μ(ρ1)τμ(ρ◦) Z μ τκ (ρ◦,ρ1) s (ρ )s (ρ◦) = τπ (ρ◦)sπ/κ(ρ1) ν even μ κ/μ 1 μ/ν ◦ Z τκ (ρ◦,ρ1) PFAFFIAN SCHUR PROCESSES AND LPP IN A HALF-QUADRANT 3041 = τπ (ρ◦)sπ/κ(ρ1) ν even sκ/ν(ρ1,ρ◦) ◦ Z τκ (ρ◦,ρ1) τ (ρ◦)s (ρ ) = π π/κ 1 . Z◦ It is not a priori obvious that U ∠ (π|κ) does deﬁne a probability distribution, but ρ◦,ρ1 this can be checked using identities (9)and(7). There may exist other solutions to (15)whereU ∠(π|μ,κ) depends on μ.We do not attempt to classify other possible choices.

3.4. The ﬁrst coordinate marginal: Last passage percolation. In this section, we relate the dynamics on Pfafﬁan Schur process constructed in Section 3.3 to half-space LPP.

DEFINITION 3.9 (Half-space geometric weight LPP). Let (ai)i≥1 be a sequence of positive real numbers and (gn,m)n≥m≥0 be a sequence of independent 9 geometric random variables with parameter anam when n ≥ m + 1 and with parameter can when n = m. We deﬁne the geometric last passage percolation time on the half-quadrant (see Figure 1), denoted G(n, m), by the recurrence for n ≥ m, max G(n − 1,m),G(n,m− 1) if n ≥ m + 1, G(n, m) = gn,m + G(n, m − 1) if n = m, with the boundary condition G(n, 0) ≡ 0.

Consider integers 0 j2 > ···>jn such that ik > jk for all k. The path k → (ik,jk) is a down right path on the half-quadrant, it is called a space-like path in the context of interacting particle systems [16, 18]. We can express the joint law of the last passage times G(ik,jk) using a Pfafﬁan Schur process.

PROPOSITION 3.10. Let c and a1,a2,... be positive real numbers. We have the equality in law (d)= (1) (n) G(i1,j1), . . . , G(in,jn) λ1 ,...,λ1 , where the sequence of partitions λ(1),...,λ(n) is distributed according the Pfafﬁan +; +; ; + | −; ; − Schur process PSP(ρ◦ ρ1 ... ρn−1 ρ1 ... ρn ) with + = ρ◦ (c, aj1+1,...,ai1 ), + = = − ρk (aik+1,...,aik+1 ) for k 1,...,n 1,

9The geometric distribution with parameter q ∈ (0, 1), denoted Geom(q), is the probability distri- k bution on Z≥0 such that if X ∼ Geom(q), P(X = k) = (1 − q)q . 3042 BAIK, BARRAQUAND, CORWIN AND SUIDAN

FIG.6. Equivalence of the two formulations in Proposition 3.10. Left: The components (i1,j1) (i2,j2) (i3,j3) λ1 ,λ1 ,λ1 of the Pfafﬁan Schur process considered in Proposition 3.10, for (i1,j1) = (3, 3), (i2,j2) = (5, 3) and (i3,j3) = (6, 2). Right: The corresponding diagram in the (3,3) (5,3) (6,2) setting of Deﬁnition 3.1. There is equality in law between λ1 ,λ1 ,λ1 on the left, and λ(1),λ(2),λ(3) on the right.

− = = − ρk (ajk+1+1,...,ajk ) for k 1,...,n 1, − = ρn (a1,...,ajn ). Equivalently, in terms of zig-zag paths, (d)= (i1,j1) (in,jn) G(i1,j1), . . . , G(in,jn) λ1 ,...,λ1 , where the sequence of partition λ(i1,j1),...,λ(in,jn) is distributed according to the Pfafﬁan Schur process indexed by a path γ ∈ going through the points (i1,j1),...,(in,jn); where vertical (resp., horizontal) edges with horizontal (resp., vertical) coordinates i −1 and i are labelled by specializations into the single variable ai , and the diagonal edge is labelled by the specialization into the variable c (See Figure 6).

(1) REMARK 3.11. Note that it is also true that μ1 has the same law as G(i1,j2), (2) μ1 has the same law as G(i2,j3), etc. but we do not use this fact.

PROOF. We begin with two lemmas explaining the action of the transition operators U and U ∠ on the ﬁrst coordinates of the partitions.

LEMMA 3.12 ([17]). Consider two specializations determined by single vari- U | ables a,b > 0 and the transition operator a,b(π ν,κ) deﬁned in (14) that updates randomly the partition μ to become π given partitions ν,κ. The ﬁrst coordinate of the partition π is such that

π1 = max(ν1,κ1) + Geom(ab). PFAFFIAN SCHUR PROCESSES AND LPP IN A HALF-QUADRANT 3043

PROOF. This lemma is a particular case of two-dimensional push-block dynamics deﬁned in [17], Section 2.6, Example (3). For two partitions λ,μ,wehave |λ|−|μ| that sλ/μ(a) = 1μ≺λa . Hence we have that |π|−|ν| |π|−|κ| U (π|ν,κ) = 1ν≺π 1κ≺π a b . Summing out π2,π3,...and using the fact that U is a stochastic transition kernel, we ﬁnd that the distribution of π1 is given by − U | = − · π1 max(ν1,κ1) (π1 ν1,κ1) (1 ab) 1π1≥max(ν1,κ1)(ab) .

LEMMA 3.13. Consider two specializations determined by single variables U ∠ | a,c > 0 and the transition operator c,a(π κ) deﬁned in (16) that updates randomly the partition μ to become π, given the partition κ. The ﬁrst coordinate of the partition π is such that

π1 = κ1 + Geom(ac).

PROOF. Similar to the proof of Lemma 3.12,wehavethat τπ (c) = π −π π −π ∠ c 1 2 c 3 4 ··· . Hence, the distribution of π1 under U (π|κ) is proportional − π1 κ1 to 1π1≥κ1 (ac) .

Fix some path γ ∈ . Let us sequentially grow the Pfaffian Schur process according to the procedure defined in Section 3.3. We need to specify the diagonal specialization and the specializations we start from on the horizontal axis. Assume that the diagonal specialization is the single variable specialization into c>0, and the specialization on the edge (i, 0) → (i − 1, 0) is the single variable specialization ai . Given how the specializations are transported on the edges of the lattice when we apply U and U ∠, this choice of initial specializations implies that for all edges of γ , vertical (resp., horizontal) edges with horizontal (resp., vertical) coordinates i −1andi are labelled by specializations into the single variable ai ,andthe diagonal edge is labelled by the specialization into the variable c (See Figure 6). It follows from Lemmas 3.12 and 3.13 that the first coordinates of the partitions λv in the sequence of Pfaffian Schur processes are such that for i>j (i,j) = (i−1,j) (i,j−1) + λ1 max λ1 ,λ1 Geom(aiaj ) and (i,i) = (i,i−1) + λ1 λ1 Geom(aic).

Hence, for any collection of vertices v1,...,vk along γ ∈ ,wehave v1 v1 (d)= λ1 ,...,λ1 G(v1),...,G(v1) , where G(v) is the last passage time to v as in Deﬁnition 3.9. 3044 BAIK, BARRAQUAND, CORWIN AND SUIDAN

= n REMARK 3.14. Consider a symmetric matrix S (gij )i,j=1 of size n whose entries gi,j are such that gi,j ∼ Geom(aiaj ) for i = j and gi,i ∼ Geom(cai).The image of S under RSK correspondence is a Young tableau10 whose shape λ is distributed according to the Pfafﬁan Schur measure PSM(c|a1,...,an). This can be deduced from [5], (7.47); see also equation (10.151) in [24]. As a consequence, this provides a very short proof that G(n, n) has the law of the λ1 marginal of PSM(c|a1,...,an). Moreover, this provides an interpretation of the other coordinates of the partition: λ1 +···+λi is the maximum over i-tuples of nonintersecting up-right paths between the points (1, 1),...,(1,i)and (n, n), . . . , (n, n − i + 1) of the sum of weights along these i paths. Note that the RSK insertion procedure also deﬁnes dynamics on (ascending) sequences of interlacing partitions, which are different from the push-block dynamics that we consider.

REMARK 3.15. Taking a Poisson limit of geometric LPP, such that the points on the diagonal also limits to a one-dimensional Poisson process, yields the Pois- sonized random involution longest increasing subsequence problem considered in [5, 6] (equivalently half-space continuous PNG with a source at the origin [38]). The limit of Proposition 3.10 shows that this corresponds to the Plancherel specialization of the Pfafﬁan Schur process.

4. Fredholm Pfaffian formulas for k-point distributions. It is shown in [21], Theorem 3.3, that if λ,¯ μ¯ is distributed according to a Pfaffian Schur process (Definition 3.1), then L ¯ := (1) − ∪···∪ (k) − (λ) 1,λi i i≥1 k,λi i i≥1 is a Pfaffian point process. In particular, for any S ≥ 1 and pairwise distinct points 11 (is,us),1≤ s ≤ S of {1,...,k}×Z we have the formal series identity P ⊂ L = ; S (17) (i1,u1),...,(iS,uS) (λ) Pf K(is,us it ,ut ) s,t=1, where K(i,u; j,v) is a 2 × 2 matrix-valued kernel K K K = 11 12 K21 K22

T with K11, K12 =−(K21) ,andK22 given by explicit formulas (see [21], Theo- rem 3.3). In Section 4.1, we ﬁrst review the formalism of Pfafﬁan point processes. In Sections 4.2 and 4.4, we will explain how [21], Theorem 3.3, applies to last passage percolation in a half-quadrant with geometric or exponential weights.

10Insertion tableau and Recording tableau are the same because S is symmetric. 11In [21], this is written as a formal identity in variables of the symmetric functions, but it can be specialized (as in Section 4.2) to numeric identities in the cases we consider. PFAFFIAN SCHUR PROCESSES AND LPP IN A HALF-QUADRANT 3045

4.1. Pfaffian point processes. We consider presently only simple12 point processes and introduce the formalism of Pfaffian point processes. Let us first start with the case of a finite state space X. A random (simple) point process on X is a probability measure on subsets X of X. The correlation functions of the point process are defined by ρ(Y)= P(X ⊂ X such that Y ⊂ X) for Y ⊂ X. k This definition implies that for a domain I1 × I2 ×···×Ik ⊂ X , ρ(x1,...,xk) (x1,...,xk)∈I1×···×Ik distinct (18) = E #{k-tuples of distinct points x1 ∈ X ∩ I1,...,xk ∈ X ∩ Ik} . Such a point process is called Pfaffian if there exists a 2×2 matrix valued |X|×|X| skew-symmetric matrix K with rows and columns parametrized by points of X, such that the correlation functions of the random point process are

(19) ρ(Y)= Pf(KY ) for any Y ⊂ X, := k ={ } where KY (K(yi,yj ))i,j=1 for Y y1,...,yk , and Pf is the Pfaffian [see (3)]. More generally, let (X,μ) be a measure space, and P (and E) be a probability measure (and expectation) on the set Conf(X) of countable and locally finite subsets (configurations) X ⊂ X.Wedefinethekth factorial moment measure on Xk by B1 ×···×Bk → E #{k-tuples of distinct points x1 ∈ X ∩ B1,...,xk ∈ X ∩ Bk} , where B1,...,Bk ⊂ X are Borel subsets. Assuming it is absolutely continuous ⊗k with respect to μ ,thek-point correlation function ρ(x1,...,xk) is the Radon– Nykodym derivative of the k-moment factorial measure with respect to μ⊗k.Asin the discrete setting, P is a Pfaffian point process if there exists a kernel

K : X × X → Skew2(R), where Skew2(R) is the set of skew-symmetric 2 × 2 matrices, such that for all Y ⊂ X,

(20) ρ(Y)= Pf(KY ). For measurable functions f : X → C, the deﬁnition of correlation functions is equivalent to ⊗k (21) E f(x1) ···f(xk) = ρ(x1,...,xk)f (x1) ···f(xk) dμ , Xk k (x1,...,xk)∈X distinct

12That is, without multiplicities. 3046 BAIK, BARRAQUAND, CORWIN AND SUIDAN and thus (20) implies that ([37], Theorem 8.2), E + = + (22) 1 f(x) Pf(J K)L2(X,f μ), x∈X whenever both sides are absolutely convergent. The RHS of (22) is a Fredholm Pfaffian, defined in Definition 2.3. We refer to [37], Section 8, and [36], Ap- pendix B, for properties of Fredholm Pfaffians. In particular, (22) implies that the gap probabilities are given by the Pfaffian formula P Y = − Y ⊂ X (23) (no point lie in ) Pf(J K)L2(Y,μ) for .

4.2. Half-space geometric weight LPP.Leta1,a2, ···∈(0, 1),andc>0such that for all i, cai < 1; and let 0 m2 > ···>mk be sequences of integers such that ni >mi for all i. Consider the Pfafﬁan Schur process indexed by the unique path in going through the points

(nk, 0), (nk,mk), (nk−1,mk), (nk−1,mk−1),...,(n1,m1), (m1,m1), (0, 0), where a horizontal edge (i − 1,j) → (i, j) [resp. a vertical edge (i, j − 1) → (i, j)] is labelled by the specialization into the single variable ai (resp., aj ), and the diagonal edge is labelled by c. In other words, we are in the same setting as in Section 3.4. In this case, it follows from [21], Theorem 3.3, that the correlation kernel of L(λ)¯ , that we will denote by Kgeo, indexed by couples of points in {1,...,k}×Z, is given by the following formulas. It is convenient to introduce the notation n m ni z − a j w − a mi 1 j 1 hgeo(z, w) := , 11 z w 1 − a z 1 − a w =1 =1 =1 =1 n m ni z − a j 1 mi 1 j w − a hgeo(z, w) := , 12 z 1 − a w 1 − a z w =1 =1 =1 =1 n m ni 1 j 1 mi z − a j w − a hgeo(z, w) := . 22 1 − a z 1 − a w z w =1 =1 =1 =1 Then the kernel is given by (z − w)hgeo(z, w) z − c w − c dz dw (24) Kgeo(i, u; j,v) = 11 , 11 (z2 − 1)(w2 − 1)(zw − 1) z w zu wv where the contours are positively oriented circles around 0 of such that for all i, 1 < |z|, |w| < 1/ai ; (z − w)hgeo(z, w) z − c dz dw Kgeo(i, u; j,v) = 12 12 (z2 − 1)w(zw − 1) z(1 − cw) zu wv (25) =− geo ; K21 (j, v i, u), PFAFFIAN SCHUR PROCESSES AND LPP IN A HALF-QUADRANT 3047

where the contours are positively-oriented circles around 0 of radius such that for all i,1< |z| < 1/ai , |w| < 1/c, 1/ai and if i ≥ j then |zw| > 1, while if i

REMARK 4.1. In the statement of Theorem 3.3 in [21], the factor (zw − 1) which appear in the denominator of the integrand in K22 is replaced by (1 − zw). This seems to be a a typo, as the proof of Theorem 3.3 in [21] suggests that the correct factor should be (zw − 1),and[27] recently conﬁrmed the correct sign.

PROPOSITION 4.2. For any g1,...,gk ∈ Z≥0, k geo (27) P G(ni,mi) ≤ gi = Pf J − K 2 ˜ , L (Dk(g1,...,gk)) i=k where ˜ Dk(g1,...,gk) = (i, x) ∈{1,...,k}×Z : x ≥ gi , and J is the matrix kernel 01 (28) J(i,u; j,v) = 1 = . (i,u) (j,v) −10

By Deﬁnition 2.3,Pf(J − Kgeo) is deﬁned by the expansion geo Pf J − K 2 ˜ L (Dk(g1,...,gk)) (29) ∞ ∞ ∞ (−1)n k = 1 + ··· Pf Kgeo(i ,x ; i ,x ) n . ! s s t t s,t=1 = n = = = n 1 i1,...,in 1 x1 gi1 xn gin

REMARK 4.3. In the special case k = 1andn1 = m1 = n, the Pfaffian rep- { (n,n) − } resentation of the correlation functions for λi i was given in [37], Corol- lary 4.3, and subsequently used in [26] to study involutions with a fixed number of fixed points. 3048 BAIK, BARRAQUAND, CORWIN AND SUIDAN

4.3. Deformation of contours. We will later need (in order to deduce the cor- √relation kernel of the exponential model from the geometric case) to set all ai to q and let q go to 1. Before taking this limit, we need to deform some of the contours used in the formulas for Kgeo and compute the residues involved. It will be much more convenient13 for the following asymptotic analysis if all integration contours in the definition of Kgeo are circles such that |zw| > 1. We do not need geo geo ≥ to transform the expressions for K11 and for K12 when i j.Whenc<1, these residues correspond to terms that arise in the Pfaffian version of Eynard–Mehta’s theorem ([21], Theorem 1.9), as in the proof of Theorem 3.3 in [21]. When c ≥ 1, we get more residues which were not present in the proof of Theorem 3.3 in [21] (recall that we make an analytic continuation to obtain the correlation kernel when c ≥ 1). These extra residues are the signature of the occurrence of a phase transition when c varies, as discovered in [6]. geo Case c<1: For K22 we write geo ; = geo ; + geo ; (30) K22 (i, u j,v) I22 (i, u j,v) R22 (i, u j,v), geo ; where I22 (i, u j,v) is the same as (26) but the contours are now positively oriented circles around 0 such that 1 < |z|, |w| < min(1/c, 1/a1, 1/a2,...);and 1 − z2 1 1 hgeo(z, 1/z) (31) Rgeo(i, u; j,v) = 22 dz. 22 z2 1 − cz 1 − c/z zu−v To see the equivalence between (26)and(30), deform the w contour so that the radius exceeds z−1. Of course, we have to subtract the residue for w = 1/z,which − geo ; is expressed as an integral in z, which equals R22 (i, u j,v). Finally, we can freely move the z contour in the two-fold integral (only after having moved the w contour) and get (30). In the case where i1: In that case, we need to take into account the residues at the poles − − geo geo of 1/(1 cz) and 1/(1 cw) in K12 and K22 . When deforming the w contour in (26), we first encounter a pole at 1/c and then a pole at 1/z. Each of these residues

13If it was not the case, we would ﬁnd issues related to the choice of limiting contours when per- forming the asymptotic analysis. PFAFFIAN SCHUR PROCESSES AND LPP IN A HALF-QUADRANT 3049

can be written as an integral in the variable z, where we may again deform the contour picking a residue at z = 1/c when necessary. Then we deform the contour for z in the two-fold integral and pick a residue at z = 1/c, which is expressed as an integral in w.Wefind geo ; = geo ; + ˆgeo ; (34) K22 (i, u j,v) I22 (i, u j,v) R22 (i, u j,v), geo where I22 is as in the c<1 case and ˆgeo ; R22 (i, u j,v) hgeo(z, 1/c) hgeo(w, 1/c) (35) =−cv 22 dz + cu 22 dw zu+1(z − c) wv+1(w − c) 2 geo 1 − z h (z, 1/z) dz − − + 22 + cu v 1hgeo(1/c, c). z2 (1 − cz)(1 − c/z) zu−v 22 Because of the particular sequence of contour deformations that we have chosen, the first integral in (35) is such that c is outside the contour, while c is inside the contour in the second and third integral in (35). We get a formula where the ˆgeo antisymmetry is apparent by deforming again the z contour and writing R22 as hgeo(z, 1/c) hgeo(w, 1/c) Rˆgeo(i, u; j,v) =−cv 22 dz + cu 22 dw 22 zu+1(z − c) wv+1(w − c) 1 − z2 hgeo(z, 1/z) dz (36) + 22 z2 (1 − cz)(1 − c/z) zu−v − u−v−1 geo + v−u−1 geo c h22 (1/c, c) c h22 (c, 1/c), where the contours are circles with radius between c and the 1/a’s in the first and second integral, and radius between 1/c and c in the third integral. geo For K12 , in the case where i

and the contours are circles around 0 such that 1 < |z| < min(1/a1, 1/a2,...). Case c = 1: When deforming contours as previously, the sequence of residues encountered is slightly different than in the case c>1, and we get geo ; = geo ; + ¯geo ; (38) K22 (i, u j,v) I22 (i, u j,v) R22 (i, u j,v), 3050 BAIK, BARRAQUAND, CORWIN AND SUIDAN

geo where I22 is as in the c<1 case and ¯geo ; R22 (i, u j,v) −1 hgeo(z, 1) 1 hgeo(w, 1) (39) = 22 dz + 22 dw 2iπ zu+1(z − 1) 2iπ wv+1(w − 1) 1 1 + z hgeo(z, 1/z) − 22 dz − hgeo(1, 1), 2iπ z2(1 − z) zu−v 22 where all contours are circles such that 1 < |z|, |w| < min(1/a1, 1/a2,...).Inthe case where i

4.4. Half-space exponential weight LPP. The following lemma is a direct consequence of the geometric to exponential weak convergence. √ √ LEMMA 4.4. Set all parameters ai ≡ q and c = q(1 + (α − 1)(q − 1)). Then, for any sequences of integers n1,...,nk and m1,...,mk such that ni ≥ mi , we have the weak convergence as q → 1, − k =⇒ k (1 q)G(ni,mi) i=1 H(ni,mi) i=1. [Recall H(n,m)from Deﬁnition 1.1 and G(n, m) from Deﬁnition 3.9.]

We will use this lemma to deduce the correlation functions for the exponential exp 2 model. Define the kernel K : ({1,...,k}×R) → Skew2(R) by ⎧ ⎪ exp ⎨⎪R (i, x; j,y) when α>1/2, exp ; = exp ; + êxp ; K (i, x j,y) I (i, x j,y) ⎪R (i, x j,y) when α<1/2, ⎩⎪ R¯exp(i, x; j,y) when α = 1/2. Recalling Definition 2.1 for integration contours, we define I exp by exp ; I11 (i, x j,y) − − (z − w)e xz yw (1 + 2z)ni (1 + 2w)nj (40) := dz dw Cπ/3 Cπ/3 + − mi − mj 1/4 1/4 4zw(z w) (1 2z) (1 2w) × (2z + 2α − 1)(2w + 2α − 1); PFAFFIAN SCHUR PROCESSES AND LPP IN A HALF-QUADRANT 3051 exp ; I12 (i, x j,y) (z − w)e−xz−yw (41) := dz dw Cπ/3 Cπ/3 + az aw 2z(z w) (1 + 2z)ni (1 + 2w)mj 2α − 1 + 2z × , (1 − 2w)nj (1 − 2z)mi 2α − 1 − 2w Cπ/3 Cπ/3 where in the definition of the contours az and aw , the real constants az,aw are chosen so that 0 0andaw <(2α − 1)/2; exp ; I22 (i, x j,y) − − (z − w)e xz yw (1 + 2z)mi (1 + 2w)mj (42) := dz dw Cπ/3 Cπ/3 z + w (1 − 2z)ni (1 − 2w)nj bz bw 1 1 × , 2α − 1 − 2z 2α − 1 − 2w where in the definition of the contours Cπ/3 and Cπ/3, the real constants b ,b bz bw z w are chosen so that 0 1/2, while we impose only bz,bw > 0whenα ≤ 1/2. exp ; = exp ; = ≥ We set R11 (i, x j,y) 0, and R12 (i, x j,y) 0wheni j, and likewise for Rêxp and R¯exp. The other entries depend on the value of α and the sign of x − y. Case α>1/2: When x>y, −| − | (1 + 2z)mi (1 − 2z)mj 2ze x y z dz Rexp(i, x; j,y) =− , 22 Cπ/3 − ni + nj − − − + az (1 2z) (1 2z) (2α 1 2z)2α 1 2z and when xy, n m 1 (1 + 2z) i (1 − 2z) j −| − | Rexp(i, x; j,y) =− e x y z dz, 12 Cπ/3 + nj − mi 2iπ 1/4 (1 2z) (1 2z) exp ; = exp ; while if xy,wehave êxp ; R22 (i, x j,y) − 1 2α y − −e 2 (1 + 2z)mi (2α)mj e xz = dz 2 (1 − 2z)ni (2 − 2α)nj 2α − 1 + 2z 3052 BAIK, BARRAQUAND, CORWIN AND SUIDAN

− 1 2α x − e 2 (1 + 2z)mj (2α)mi e yz (43) + dz 2 (1 − 2z)nj (2 − 2α)ni 2α − 1 + 2z m m (1 + 2z) i (1 − 2z) j 1 1 −| − | − 2ze x y z dz Cπ/3 − ni + nj − − − + az (1 2z) (1 2z) 2α 1 2z 2α 1 2z − − (x−y)1 2α (y−x)1 2α e 2 (2α)mi (2 − 2α)mj e 2 (2α)mj (2 − 2α)mi − + , 4 (2 − 2α)ni (2α)nj 4 (2 − 2α)nj (2α)ni where the contours in the two first integrals pass to the right of (1 − 2α)/2. êxp ; = When xy, − 1 2α y − −e 2 (1 + 2z)mi (2α)mj e xz Rêxp(i, x; j,y) = dz 22 Cπ/3 − ni − nj − + 2 az (1 2z) (2 2α) 2α 1 2z − 1 2α x − e 2 (1 + 2z)mj (2α)mi e yz + dz Cπ/3 − nj − ni − + 2 az (1 2z) (2 2α) 2α 1 2z (44) + mi − mj − (1 2z) (1 2z) Cπ/3 − ni + nj az (1 2z) (1 2z)

1 1 −| − | × 2ze x y z dz, 2α − 1 − 2z 2α − 1 + 2z 2α−1 1−2α where 2 y n m (1 + 2z) i (1 − 2z) j −| − | (45) Rêxp(i, x; j,y) =− e x y z dz, 12 Cπ/3 + nj − mi 1/4 (1 2z) (1 2z) êxp ; = êxp ; while if x1/c in the contours for I12 in the case c>1. Case α = 1/2: When x>y, − − (1 + 2z)mi e xz (1 + 2z)mj e yz R¯exp(i, x; j,y) =− dz + dz 22 Cπ/3 − ni Cπ/3 − nj 1/4 (1 2z) 4z 1/4 (1 2z) 4z −| − | (1 + 2z)mi (1 − 2z)mj e x y z dz 1 + − , Cπ/3 − ni + nj 1/4 (1 2z) (1 2z) 2z 4 PFAFFIAN SCHUR PROCESSES AND LPP IN A HALF-QUADRANT 3053 ¯exp ; = with a modification of the last two terms when xy ¯exp ; R12 (i, x j,y) n m (1 + 2z) i (1 − 2z) j −| − | =− e x y z dz, Cπ/3 + nj − mi 1/4 (1 2z) (1 2z) ¯exp ; = ¯exp ; while if x

PROOF OF PROPOSITION 1.6. By Lemma 4.4, the passage times {H(ni,mi)}i are the limits when q goes to 1 of the passage times {G(ni,mi)}i , and we know the probability distribution function of {G(ni,mi)}i from (27). Thus, the proof proceeds with two steps: (1) Take the asymptotics of the correlation kernel Kgeo { } involved in the√ probability√ distribution of G(ni,mi) i (Section 4.2), under the scalings ai ≡ q and c = q(1 + (α − 1)(q − 1)), x = (1 − q)u, y = (1 − q)v; (2) Deduce the asymptotic behavior of Pf(J + Kgeo) from the pointwise asymptotics of Kgeo. Using the weak convergence of Lemma 4.4, we get the distribution of {H(ni,mi)}i . Step (1): We use Laplace’s method to take asymptotics of Kgeo(i, u; j,v) under geo the scalings above. Let us provide a detailed proof for K11 . The arguments are geo geo quite similar for K12 and K22 .Wehave geo ; K11 (i, u j,v) √ √ (z − w)(z − c)(w − c) (z − q)ni (w − q)nj (46) = √ √ C2 (z2 − 1)(w2 − 1)(zw − 1) (1 − z q)mi (1 − w q)mj

× dz dw + + , zni 1wnj 1zuwv − C 1+q 1/2 where the contour can be chosen as a circle around 0 with radius 2 . Under the scalings considered, the dominant term in the integrand is

1 = 1 − − zuwv z(1−q) 1xw(1−q) 1y − = exp (1 − q) 1 −x log(z) − y log(w) . Using Cauchy’s theorem, we can deform the integration contours as long as we do not cross any pole. Hence, we can deform the contour C to be the contour C< 1+q−1/2 depicted in Figure 7. It is constituted of two rays departing 2 with angles ±π/3 in a neighborhood of 1 of size ε (for some small ε>0),andtherestofthe 1+q−1/2 contour is given by an arc of a circle centered at 0. Its radius R> 2 is chosen 1+q−1/2 so that the circle intersects with the endpoints of the two rays departing 2 with angles ±π/3. Along the contour C<, Re[− log(z)] attains its maximum at 3054 BAIK, BARRAQUAND, CORWIN AND SUIDAN

FIG.7. The deformed contour C< (thick black line) and the contour C (dashed line). On the right, zoom on the portion of the contour which contributes to the limit.

1+q−1/2 2 . Thus, the contribution of integration along the portion of contour which is outside the ε-box around 1 is negligible in the limit. More precisely, it has a −d (1−q)−1 contribution of order O(e ε ) for some dε > 0. Now we estimate the contribution of the integration in the neighborhood of 1. Let us rescale the variables around 1 by setting z = 1 +˜z(1 − q) and w = 1 + w(˜ 1 − q). Cauchy’s theorem provides us some freedom in the choice of contours, and we can assume that after the change of variables, the integration contours are given by two rays of length ε(1 − q)−1 departing 1/4 in the directions π/3and −π/3. When q → 1, we have the point-wise limits √ ˜ z − q z − c zu −→ ezx, √ −→ 1 + 2z,˜ √ −→ − 1 + 2z˜ + 2α, 1 − q 1 − q √ ˜ 1 − z q 1 − zc wv −→ ewy, √ −→ 1 − 2z,˜ √ −→ − 1 − 2z˜ + 2α. 1 − q 1 − q

Using Taylor expansions to a high enough order, the integral in (46)converges to the integral of the pointwise limit of the integrand in (46), modulo a O(ε) error, uniformly in q. The kind of estimates needed to control the error are given with more details in a similar situation in Section 5.2 [see in particular equations (71)and(73)]. Moreover, since the limiting integrand has exponential decay in direction ±π/3, one can replace the finite integration contours by infinite contours going to ∞eiπ/3 and ∞e−iπ/3 with an error going to 0 as q goes to 1. Since the size ε of the box around 1 (on which we have restricted the integration) can be taken as small as we want, we finally get PFAFFIAN SCHUR PROCESSES AND LPP IN A HALF-QUADRANT 3055

that geo : K11 (i, u j,v) √ + − − + (1 − q)ni nj mi mj 2 z˜ −˜w − ˜− ˜ −−−→ dz˜ dw˜ e xz yw q→1 Cπ/3 Cπ/3 ˜ ˜ ˜ +˜ 1/4 1/4 4zw(z w) (1 + 2z)˜ ni (1 + 2˜w)nj × (2z˜ + 2α − 1)(2w˜ + 2α − 1). (1 − 2z)˜ mi (1 − 2w)˜ mj Likewise, we obtain geo : I12 (i, u j,v) √ + − − (1 − q)(1 − q)ni mj nj mi z˜ −˜w − ˜− ˜ −−−→ dz˜ dw˜ e xz yw → Cπ/3 Cπ/3 ˜ ˜ +˜ q 1 az aw 2z(z w) (1 + 2z)˜ ni (1 + 2w)˜ mj 2α − 1 + 2z˜ × , (1 − 2w)˜ nj (1 − 2z)˜ mi 2α − 1 − 2w˜ Cπ/3 Cπ/3 where the new contours az and aw are chosen as in (41). In the case where = Cπ/3 α<1/2orα 1/2, the contour aw is not exactly the limit of the contour in the geo = − definition of I12 in (32), but we can exclude the pole at w (2α 1)/2[sothat − êxp ¯exp aw <(2α 1)/2] and remove the corresponding residue in R12 and R12 (see Remark 4.5). Thus, we have that geo ; R12 (i, u j,v) exp (47) √ + − − −−−→ R (i, x; jy), (1 − q)(1 − q)ni mj nj mi q→1 12 ˆgeo ¯geo and likewise for R12 and R12 (modulo the residue that is reincorporated in exp I12 ). Convergence (47) needs some justification though: for x>y,weuse the exact same arguments as above, since the factor 1/(zu−v) ≈ e−(x−y)z˜ in (33) has exponential decay along the tails of the contour. For xy. This shows that geo ; K12 (i, u j,v) exp √ + − − −−−→ K (i, x; jy). (1 − q)(1 − q)ni mj nj mi q→1 12 Using very similar arguments, we find that geo ; I22 (i, u j,v) √ + − − − (1 − q)2(1 − q)mi mj ni nj 2 z˜ −˜w − ˜− ˜ −−−→ dz˜ dw˜ e xz yw q→1 Cπ/3 Cπ/3 z˜ +˜w bz bz 3056 BAIK, BARRAQUAND, CORWIN AND SUIDAN

+ ˜ mi + ˜ mj × (1 2z) (1 2w) (1 − 2z)˜ mi (1 − 2w)˜ mj 1 1 × , 2α − 1 − 2z˜ 2α − 1 − 2w˜ where Cπ/3 and Cπ/3 are chosen as in (42), and bz bw Rgeo(i, u; j,v) 22 −−−→ exp ; √ + − − − R (i, x jy), (1 − q)2(1 − q)mi mj ni nj 2 q→1 22 ˆgeo ¯geo and similar limits hold for R22 and R22 . √ + − − + Note that when taking the Pfaffian, the factors (1− q)ni nj mi mj 2 and (1− √ + − − − q)mi mj ni nj 2 cancel out each other. This is because one can multiply the √ + − − − columns with odd index by (1 − q)mi mj ni nj 2 and then the rows with even √ + − − + index by (1 − q)ni nj mi mj 2, without changing the value of the Pfaffian.14 In the end, we find that for all integer n ≥ 1, positive real variables x1,...,xn,and integers 1 ≤ i1,...,in ≤ k, [ geo − −1 ; − −1 ]n Pf K (is,(1 q) xs it ,(1 q) xt ) s,t=1 exp n −−−→ Pf K (is,xs; it ,xt ) = . (1 − q)n q→1 s,t 1 Step (2): We need to prove that k +∞ +∞ ··· geo ; n Pf K (is,us it ,ut ) s,t=1 − − i ,...,in=1 = − 1 = − 1 1 u1 gi1 (1 q) un gin (1 q) (48) k +∞ +∞ −−−→ ··· Pf Kexp(i ,x ; i ,x ) n dx ··· dx . → s s t t s,t=1 1 k q 1 gi gi i1,...,in=1 1 n For that purpose, we need to control the asymptotic behavior of the kernel Kgeo.

1 1−2α LEMMA 4.6. There exists positive constants C>0 and 2 >c1 >c2 > 2 such that uniformly for q in a neighborhood of 1, for all x,y > 0, Kgeo(i, (1 − q)−1x; j,(1 − q)−1y) 11 −c1(x+y) (49) √ + − − ≤ Ce , (1 − q)ni nj mi mj Kgeo(i, (1 − q)−1x; j,(1 − q)−1y) 12 −xc1+yc2 (50) √ + − − ≤ Ce , (1 − q)ni nj mi mj Kgeo(i, (1 − q)−1x; j,(1 − q)−1y) 22 xc2+yc2 (51) √ + − − ≤ Ce . (1 − q)ni nj mi mj

14Otherwise said, we conjugate the kernel by a diagonal kernel with determinant 1. PFAFFIAN SCHUR PROCESSES AND LPP IN A HALF-QUADRANT 3057

PROOF. The arguments are similar for all integrals involved. It amounts to justify that we can restrict the integration on a region on the complex plane with real part greater than c1 or −c2. Let us ﬁrst see how it works for the bound (49). We start with (46) as in Step (1). The contribution of integration along the circular parts of C< can be bounded by a constant for q in a neighborhood of 1. Next, we perform the change of variables z = 1 +˜z(1 − q) and w = 1 +˜w(1 − q) as in Step (1) and observe that all factors in the integrand are bounded for q in a neighborhood of 1, except the factor 1/(zuwv) which behaves as

1 − − = exp −x(1 − q) 1 log 1 +˜z(1 − q) − y(1 − q) 1 log 1 +˜w(1 − q) zuwv − ˜− ˜ ≈ e xz yw.

Since we can deform the contours so that Re[˜z] and Re[˜w] are greater than c1 (for some c1 < 1/2), we get the desired bound for K11. Strictly speaking, Cauchy’s theorem allows to deform the contour C< so that the real part stays above c1 along the two rays, and then on cuts the circular part which has a negligible contribution. Similarly, we ﬁnd that I geo(i, (1 − q)−1x; j,(1 − q)−1y) 12 −xc1+yc2 √ + − − ≤ Ce (1 − q)ni nj mi mj

by restricting the integration on contours such that Re[˜z] >c1 and Re[˜w] > −c2. 2α−1 When deforming the contour for w, we have to take into account the pole at 2 . 1−2α 1 Since for any α>0, 2 < 2 , we can always ﬁnd suitable constants c1,c2 > 0 1 1−2α such that 2 >c1 >c2 > 2 . By the same kind of arguments, Rˆgeo(i, (1 − q)−1x; j,(1 − q)−1y) 12 −c|x−y| −xc1+yc2 √ + − − ≤ Ce

Hence using Lemma 2.5, − − − − +···+ geo − 1 ; − 1 n ≤ n/2 n (c1 c2)(x1 xn) Pf K is,(1 q) xs it ,(1 q) xt s,t=1 (2n) C e 3058 BAIK, BARRAQUAND, CORWIN AND SUIDAN and (48) follows by dominated convergence. Dominated convergence also proves the convergence of Fredholm Pfafﬁan series expansions, so that − geo −−−→ − exp (52) Pf J K 2 D˜ − −1 − −1 Pf J K L2 D . ( k(h1(1 q) ,...,hk(1 q) )) q→1 ( k(h1,...,hk)) Finally, Proposition 4.4 implies that k ! k xi P G(ni,mi) ≤ −−−→ P H(ni,mi) ≤ xi , 1 − q q→1 i=1 i=1 which, together with (52), concludes the proof.

5. Convergence to the GSE Tracy–Widom distribution. This section is devoted to the proof of the first part of Theorem 1.3, and thus we assume that α>1/2. By Proposition 1.6, the probability P(H (n, n) < x) can be expressed as the Fredholm Pfaffian of the kernel Kexp. The most natural would be to prove that under the scalings considered, Kexp converges pointwise to KGSE, and that the convergence of the Fredholm Pfaffian holds as well, using estimates to control the absolute convergence of the series expansion. Unfortunately, this approach cannot work here because Kexp does not converge to KGSE. The next remark provides a heuristic explanation.

REMARK 5.1. We expect the kernel Kexp(1,x; 1,y)to be the Pfaffian correla- (n) := { (n) ··· (n)} tion kernel of some simple Pfaffian point process, say X X1 > >Xn . On the other hand, KGSE(x, y) is the correlation kernel of a Pfaffian point process χ1 >χ2 > ··· having the same law as the first eigenvalues of a matrix from the GSE in the large size limit. (n) (n) +···+ (n) In light of Remark 3.14, the points Xi are such that X1 Xi has the same law as the maximal weight of a i-tuple of nonintersecting up-right paths in a symmetric exponential environment with weights E(α) on the diagonal and E (n) (1) off-diagonal [in particular X1 has the same distribution as H(n, n) since the half-space LPP and symmetrized LPP are equivalent]. When α is large enough, it (n) is clear that the optimal paths will seldom visit the diagonal, so that X1 and (n) 1/3 X2 will have (under n scaling) the same scaling limit. More precisely, we (n) (n) (n) expect that X2i−1 and X2i will both converge to χi , so that the point process X converges as n goes to infinity to a nonsimple point process where each point has multiplicity 2. GSE Let us denote by ρ the correlation function of the point process χ1,χ2,..., (n)− and let ρ(n) be the correlation function of the rescaled point process X 4n .By 24/3n1/3 the characterization of correlation functions in (21), we expect that for distinct points x1,...,xk, (n) k GSE ρ (x1,...,xk) −−−→ 2 ρ (x1,...,xk), n→∞ PFAFFIAN SCHUR PROCESSES AND LPP IN A HALF-QUADRANT 3059

(n) and a singularity should appear in the limit of ρ (x1,...,xk) for k>1inthe neighborhood of xi = xj .

5.1. A formal representation of the GSE distribution. We introduce a kernel which is a scaling limit of Kexp and whose Fredholm Pfaffian corresponds to the GSE distribution. However, it is not the Pfaffian correlation kernel of the GSE point process. This kernel is distribution valued, and we show in Proposition 5.2 that its Fredholm Pfaffian is the same as that of KGSE, by expanding and reordering terms in the Fredholm Pfaffian. In fact, to avoid dealing with distribution valued kernels, we later prove an approximate prelimiting version of this result as Proposition 5.7. Let us first recall that by Lemma 2.7,thekernelKGSE has the form A(x,y) −∂ A(x,y) KGSE(x, y) = y , −∂xA(x,y) ∂x∂yA(x,y) GSE where A(x,y) is the smooth and antisymmetric kernel K11 . We introduce the kernel ∞ A(x,y) −2∂yA(x,y) K (x, y) = , −2∂xA(x,y) 4∂x∂yA(x,y) + δ (x, y) R2 where δ is a distribution on such that (53) f(x,y)δ (x, y) dx dy = ∂yf(x,y) − ∂xf(x,y) |y=x dx, for smooth and compactly supported test functions f . The quantity ∞ k (54) Pf K (xi,xj ) = dx1 ··· dxk Rk i,j 1

makes perfect sense because all terms of the form δ (xi,xj ) are integrated against smooth functions of xi,xj with sufficient decay. Expanding the Pfaffian and using the definition of δ ,(54) collapses into a sum of j-fold integrals for j = k down to k −k/2, where the integrands are smooth functions. We can then form the Fredholm Pfaffian of K∞, and we will show that the Fredholm Pfaffian series expansion is absolutely convergent so that we can reorder terms, and ultimately recognize the Fredholm Pfaffian of another kernel.

2 PROPOSITION 5.2. Let A : R → Skew2(R) be a kernel of the form A(x,y) −∂ A(x,y) A(x, y) = y , −∂xA(x,y) ∂x∂yA(x,y) where A is smooth, antisymmetric, and A satisﬁes the decay hypotheses of Lemma 2.5. Let B be the kernel A(x,y) −2∂yA(x,y) B(x, y) = . −2∂xA(x,y) 4∂x∂yA(x,y) + δ (x, y) 3060 BAIK, BARRAQUAND, CORWIN AND SUIDAN

Then for any x ∈ R, [ − ] = [ − ] (55) Pf J A L2(x,+∞) Pf J B L2(x,+∞). In particular, − GSE = − ∞ Pf J K L2(x,+∞) Pf J K L2(x,+∞).

PROOF OF PROPOSITION 5.2. In order to simplify the notation in this proof, we assume that all integrals are performed over the domain (x, +∞). Denoting = ··· m bm dx1 dxmPf B(xi,xj ) i,j=1 and = ··· k ak dx1 dxkPf A(xi,xj ) i,j=1, we have ∞ ∞ (−1)k (−1)m Pf[J − A]= a and Pf[J − B]= b . k! k m! m k=0 m=0

In order to analyze the quantity bm, we will use the following deﬁnition of the Pfafﬁan, valid for any antisymmetric matrix M of size 2m: [ ]= ··· (56) Pf M ε(α)Mi1,j1 Mim,jm , α={(i1,j1),...,(im,jm)} where the sum runs over partitions into pairs of the set {1,...,2m}, in the form α = (i1,j1),...,(im,jm) with i1 < ···

The key fact toward the proof of the proposition is that k − k k ( 1) (57) b + = a . k i k! k i=0 This corresponds to the fact that k-fold integral terms match in the expansion of both sides of (55). Then we can write ∞ − m ∞ m ∞ k ∞ − k ( 1) k k ( 1) (58) b = b = b + = a , m! m m k i k! k m=0 m=0 k=m/2 k=0 i=0 k=0 which is what we wanted to prove. In order to reorder terms in the third equality k in (58) (using Fubini’s theorem), we must check that the double series bk+i is absolutely convergent. Using Lemma 2.5, it is enough to check that ∞ k 1 k − 2k 1(2k)k/2Ck, k! i k=0 i=0 is absolutely convergent, which is clearly the case [C is some constant depending on A(x,y)]. The rest of this proof is devoted to the proof of (57). We will actually prove that − k k ( 1) k k−i i b + = 2 (−1) a k i k! i k from which (57) is deduced easily by summing over i: k − k k − k k ( 1) k k−i i ( 1) b + = 2 (−1) a = a . k i k! i k k! k i=0 i=0 The matrix B can be seen as the sum of a smooth matrix with good decay properties and another matrix whose entries are generalized functions of the type

δ (xi,xj ). The following minor summation formula will be useful.

LEMMA 5.3 ([39] Lemma 4.2). Let M,N be skew symmetric matrices of size n for some even n. Then |I|−#I/2 Pf[M + N]= (−1) Pf[MI ]Pf[NI c ], I⊂{1,...,n};#I even where #I denotes the cardinality of I, |I| the sum of its elements, and I c its com- plement in {1,...,n}. For I ⊂{1,...,n}, MI denotes the matrix M restricted to rows and columns indexed by elements of I.

≥ k = Let us ﬁx some k 1 and compute bk+i for i 1tok. It is instructive to consider ﬁrst the case i = 1. 3062 BAIK, BARRAQUAND, CORWIN AND SUIDAN

k k Computation of bk+1: By deﬁnition bk+1 corresponds to the terms of degree 1 in δ in the Pfafﬁan expansion of − k+1 k+1 ( 1) A11(xi,xj ) 2A12(xi,xj ) + 00 + ! Pf (k 1) 2A21(xi,xj ) 4A22(xi,xj ) 0 δ (xi,xj ) i,j=1 after taking the integral over x1,...,xk+1. Using Lemma 5.3, we can write − k+1 k ( 1) b + = dx ··· dx + k 1 (k + 1)! 1 k 1 (59) " # × − 2i+2j−1 2i,2j δ (xi,xj )( 1) Pf Ak+1 , 1≤i

2"i,2#j where An denotes the 2(n − 1) × 2(n − 1) dimensional matrix which is the result of removing columns and rows 2i and 2j from A (x ,x ) 2A (x ,x ) n 11 a b 12 a b . 2A21(xa,xb) 4A22(xa,xb) a,b=1 By using operations on rows and columns, we ﬁnd that all terms in the summation in the RHS of (59) are equal modulo a permutation of the xi . Since we integrate the xi over a symmetric domain, these permutations are inconsequential so that + − k 1 + # k ( 1) k(k 1) 2k,2(k+1) b + =− dx ··· dx + δ (x ,x + )Pf A . k 1 2(k + 1)! 1 k 1 k k 1 k+1

LEMMA 5.4. The action of δ is such that # ··· 2k,2(k+1) =− k dx1 dxk+1δ (xk,xk+1)Pf Ak+1 2 ak.

2#k,2(k+1) PROOF. We will use the shorthand notation Mk+1 for the matrix Ak+1 , 2#k,2(k+1) and Mk for the matrix formed from the matrix Ak+1 after swapping the column and row of indices 2k −1and2k. The notation Mj for j = k,k+1 means that the column and row where the variable xj appears is on the rightmost/bottommost # # [ ]=− [ 2k,2(k+1)] [ ]= [ 2k2(k+1)] position. We have Pf Mk Pf Ak+1 and Pf Mk+1 Pf Ak+1 .Then # ··· 2k,2(k+1) dx1 dxk+1δ (xk,xk+1)Pf Ak+1 = ··· [ ]| + [ ]| dx1 dxk ∂xk Pf Mk xk+1=xk ∂xk+1 Pf Mk+1 xk+1=xk . Moreover, expanding the Pfafﬁan as in (29)or(56), we see that [ ] | = | ∂xk Pf Mk xk+1=xk Pf (∂xk Mk) xk+1=xk PFAFFIAN SCHUR PROCESSES AND LPP IN A HALF-QUADRANT 3063

and [ ] | = | ∂xk+1 Pf Mk+1 xk+1=xk Pf (∂xk+1 Mk+1) xk+1=xk . =− Using the differential relations ∂xk A11(x1,xk) A12(x1,xk), ∂xk A11(xk, =− =− = x1) A21(xk,x1) and ∂xk A21(x1,xk) A22(x1,xk), ∂xk A12(xk,x1) −A22(xk,x1), we see that k−1 k | = =− Pf (∂xk Mk) xk+1 xk 2 Pf A(xi,xj ) i,j=1, and we obtain similarly that k−1 k + | = =− Pf (∂xk+1 Mk 1) xk+1 xk 2 Pf A(xi,xj ) i,j=1. Thus, # ··· 2k,2(k+1) dx1 dxk+1δ (xk,xk+1)Pf Ak+1 =− k ··· k =− k 2 dx1 dxkPf A(xi,xj ) i,j=1 2 ak.

Thus, Lemma 5.4 shows that − k+1 k ( 1) k−1 b + = k2 a . k 1 k! k k k Computation of bk+i in general: By deﬁnition bk+i corresponds to the terms of degree i in δ in the Pfafﬁan expansion of − k+i k+i ( 1) A11(xi,xj ) 2A12(xi,xj ) + 00 + ! Pf (k i) 2A21(xi,xj ) 4A22(xi,xj ) 0 δ (xi,xj ) i,j=1

after taking the integral over x1,...,xk+i. Using Lemma 5.3, − k+i k ( 1) 2i b + = dx ··· dx + Pf δ (x ,x ) k i (k + i)! 1 k i jr js r,s=1 1≤j1<···

[ ]k+i The Pfafﬁan Pf δ (xr ,xs) r,s=k−i+1 can be expanded into products of δ acting on disjoint pairs of variables. Each term in the expansion will give the same contribution after we have integrated over the xj s, in the sense of the following.

LEMMA 5.5. For any partition into pairs of the set {k − i + 1,...,k+ i}, in the form α ={(r1,s1),...,(ri,si)} with r1 < ···

(61) = ε(α) dx1 ··· dxk+i δ (xk−i+2j−1,xk−i+2j ) j=1 × 2(k−i+1),...,2(k+i) Pf Ak+i .

PROOF. In order to match both sides of (61), we make a change of variables so that the δ factors exactly match. This does not change the value of the integral. Then we have to exchange rows/columns in the Pfafﬁan. Each time that we exchange two adjacent rows/columns,15 we factor out a (−1), hence the signature prefactor on the RHS of (61).

LEMMA 5.6. The action of δ is such that i ··· 2(k−i+1),...,2(k+i) dx1 dxk+i δ (xk−i+2j−1,xk−i+2j )Pf Ak+i j=1 (62) i k = (−1) 2 ak.

PROOF. We adapt the proof of Lemma 5.4.Forj1 < ···

2(k−i+1),...,2(k+i) we denote by Mj1,...,ji the matrix obtained from Ak+i where for all − + r, one has moved the column and row where xjr appears to position 2(k i r).

With this deﬁnition, if jr is even then the column and row where xjr appears do not move, and if jr is odd then the column and row where xjr appears is swapped with the right/bottom neighbor. Hence, we have that

− + + +···+ k i 1,...,k i = − j1 ji [ ] Pf Ak+i ( 1) Pf Mj1,...,ji .

15Recall that in Pfafﬁan calculus, one always exchanges rows and columns simultaneously, to preserve the antisymmetry of the matrix. PFAFFIAN SCHUR PROCESSES AND LPP IN A HALF-QUADRANT 3065

By the deﬁnition of δ , i ··· 2(k−i+1),...,2(k+i) dx1 dxk+i δ (xk−i+2j−1,xk−i+2j )Pf Ak+i j=1 i = ··· − (63) dx1 dxk (∂xk−i+2j ∂xk−i+2j−1 ) j=1 × 2(k−i+1),...,2(k+i) | Pf Ak+i xk+i =···=xk+1=xk . We can develop the product of partial derivatives so that i (63) = dx ··· dx (−1)j ∂ 1 k xj j1,...,ji =1 × 2(k−i+1,...,2(k+i) | Pf Ak+i xk+i =···=xk+1=xk = dx ··· dx ∂ ···∂ Pf[M ]| + =···= + = , 1 k xj1 xji j1,...,ji xk i xk 1 xk j1,...,ji where the sums runs over all sequences j1 < ···

! There are (2i) terms in the expansion of the Pfafﬁan of the 2i × 2i matrix 2i i! k+i (δ (xr ,xs))r,s=k−i+1. Combining Lemmas 5.6 and 5.5,wehave − k+i + ! − k k ( 1) k i (2i) k ( 1) k k−i i b + = 2 a = 2 (−1) a . k i (k + i)! 2i 2ii! k k! i k

Proposition 5.2 suggests that if the kernel Kexp converges to K∞ under some exp scalings, then the Fredholm Pfaffian of K should converge to FGSE.However, the kernel K∞ is distribution valued, and we do not know an appropriate notion of convergence to distribution-valued kernels. The following provides an approxi- mative version of Proposition 5.2 that provides a notion of convergence sufficient for the Fredholm Pfaffian to converge. 3066 BAIK, BARRAQUAND, CORWIN AND SUIDAN

PROPOSITION 5.7. Let (An(x, y))n∈Z≥0 be a family of smooth and antisym- ∈ Z r s metric kernels such that for all r, s ≥0, ∂x ∂yAn(x, y) has exponential decay at inﬁnity (in any direction of R2), uniformly for all n. Let An(x, y) −∂yAn(x, y) An(x, y) := −∂xAn(x, y) ∂x∂yAn(x, y) and − := An(x, y) 2∂yAn(x, y) Bn(x, y) − + 2∂xAn(x, y) 4∂x∂yAn(x, y) δn(x, y) 2 be two families of kernels (R) → Skew2(R) such that: 1. The family of kernels An(x, y) −2∂yAn(x, y) −2∂xAn(x, y) 4∂x∂yAn(x, y) satisfy, uniformly in n, the hypotheses of Lemma 2.5. 2. The kernel An(x, y) converges pointwise to A(x,y) −∂ A(x,y) A := y , −∂xA(x,y) ∂x∂yA(x,y) 2 a smooth kernel R → Skew2(R) (necessarily satisfying the hypotheses of Lemma 2.5). : R2 → R r s 3. For any smooth function f such that ∂x ∂yf(x,y) have exponential decay at inﬁnity for all r, s, we have − − | δn(x, y)f (x, y) dx dy ∂yf(x,y) ∂xf(x,y) x=y dx

=: cn(f ) −−−→ 0. n→∞ Then for all x ∈ R,

Pf[J − Bn]L2 ∞ −−−→ Pf[J − A]L2 ∞ . (x, ) n→∞ (x, )

PROOF. We will follow the proof of Proposition 5.2, and make approximations when necessary. Let = ··· m bm(n) dx1 dxmPf Bn(xi,xj ) i,j=1, = ··· m am(n) dx1 dxmPf An(xi,xj ) i,j=1,

k (−1)m and denote by bm(n) the part of the expansion of m! bm(n) involving only terms − of degree m k in δn [as in (60)]. We need to justify the approximation k − k k ( 1) b + (n) ≈ a (n), k i k! k i=0 PFAFFIAN SCHUR PROCESSES AND LPP IN A HALF-QUADRANT 3067

and show that the error is summable and goes to zero. Let us replace A, B by An, Bn in the proof of Proposition 5.2. The only part of the proof which breaks down is when we apply the deﬁnition of δ . However, using the hypothesis (3), we see that replacing δ by δ in Lemma 5.6 results in an error bounded by n i i 2 (cn) ak(n) ≥ = ˜k for i 1 and the error is zero when i 0. Let bk+i(n) be the quantity having the k same expression as bk+i(n) where all occurrences of δn are replaced by δ .Inother ˜k terms, b + (n) is given by (60) where the kernel is An instead of A.Wehave k i k ˜k 1 k k−i i i b + (n) − b + (n) ≤ 2 2 (c ) a (n) . k i k i k! i n k On one hand, ∞ ∞ (−1)m m Pf[J − B ]= b (n) = bk (n), n m! m m m=0 m=0 k=m/2 while on the other hand, ∞ − k ∞ k ∞ m ( 1) ˜k ˜k Pf[J − A ]= a = b + = b (n), n k! k k i m k=0 k=0 i=0 m=0 k=m/2 wherewehaveused(57) from the proof of Proposition 5.2 in the second equality and Fubini’s theorem in the third equality [which we can do by hypothesis (1) and Lemma 2.5]. Thus, ∞ m 1 k − Pf[J − B ]−Pf[J − A ] ≤ 2k(c )m ka (n) n n k! m − k n k m=0 k=m/2 ∞ k 1 k = 2k(c )ia (n) k! i n k k=0 i=1 ∞ 2k = a (n) (1 + c )k − 1 . k! k n k=0 Using the bound, k k k (1 + cn) − 1 ≤ k log(1 + cn)(1 + cn) ≤ kcn(1 + cn) , we ﬁnd that ∞ k 2 k Pf[J − Bn]−Pf[J − An] ≤ ak(n) kcn(1 + cn) −−−→ 0. k! n→∞ k=0 Moreover, using the pointwise convergence of hypothesis (2) and the decay hypothesis (1) along with Lemma 2.5, dominated convergence shows that

Pf[J − An]−−−→ Pf[J − A]. n→∞ 3068 BAIK, BARRAQUAND, CORWIN AND SUIDAN

5.2. Asymptotic analysis of the kernel Kexp. Recall that we have assumed that α>1/2 (as in the first part of Theorem 1.3). Let us first focus on pointwise convergence of Kexp(1,x; 1,y) as n = m goes to infinity. In the following, we write simply Kexp(x; y) instead of Kexp(1,x; 1,y). We expect that as n goes to infinity, H(n, n) is asymptotically equivalent to hn for some constant h, and fluctuates around hn in the n1/3 scale. Thus, it is natural to study the correlation kernel at a point (64) r(x, y) := nh + n1/3σx,nh+ n1/3σy , where the constants h, σ will be fixed later. We introduce the rescaled kernel ⎛ ⎞ 1 exp 1/3 exp exp,n ⎝ K11 r(x, y) σn K12 r(x, y) ⎠ K (x, y) := (2α − 1)2 . 1/3 exp 2 2/3 − 2 exp σn K21 r(x, y) σ n (2α 1) K22 r(x, y) By a change of variables in the Fredholm Pfaffian expansion of Kexp and a conjugation of the kernel preserving the Pfaffian, we get using Proposition 1.6 that H(n, n) − hn P ≤ y = Pf J − Kexp,n . σn1/3 L2(y,∞) Moreover, we write Kexp,n(x, y) = I exp,n(x, y) + Rexp,n(x, y), exp,n according to the decomposition made in Section 4.4.ThekernelI11 can be rewritten as z − w (2z + 2α − 1)(2w + 2α − 1) I exp,n(x, y) = dz dw 11 Cπ/3 Cπ/3 + − 2 1/4 1/4 4zw(z w) (2α 1) (65) × exp n f(z)+ f(w) − n1/3σ(xz+ yw) , where (66) f(z)=−hz + log(1 + 2z) − log(1 − 2z). Setting h = 4, we find that f (0) = f (0) = 0. Setting σ = 24/3, Taylor expansion of f around zero yields σ 3 (67) f(z)= z3 + O z4 . 3 exp,n Similarly, the kernel I12 can be rewritten as z − w 2α − 1 + 2z I exp,n(x, y) = σn1/3 dz dw 12 Cπ/3 Cπ/3 + − − 1/4 0 2z(z w) 2α 1 2w (68) × exp n f(z)+ f(w) − n1/3σ(zx+ wy) . PFAFFIAN SCHUR PROCESSES AND LPP IN A HALF-QUADRANT 3069

Here, the assumption that α>1/2 matters in the choice of contours, so that the = − exp,n poles for w (2α 1)/2 lie on the right of the contour. The kernel I22 can be rewritten as z − w (2α − 1) (2α − 1) I exp,n(x, y) = σ 2n2/3 dz dw 22 Cπ/3 Cπ/3 + − − − − 1/4 1/4 z w 2α 1 2z 2α 1 2w × exp n f(z)+ f(w) − n1/3σ(zx+ wy) . exp,n exp,n = exp,n = exp,n = The kernel R is such that R11 R12 R21 0and − 2 − exp,n 1/3 2α 1 −|x−y|σn1/3 2α 1 R (x, y) = sgn(y − x) σn e 2 . 22 2

PROPOSITION 5.8. For α>1/2 and σ = 24/3, we have

3 3 (z − w)ez /3+w /3−xz−yw I exp,n(x, y) −−−→KGSE(x, y) = dz dw , 11 n→∞ 11 Cπ/3 Cπ/3 + 1 1 4zw(z w) 3 3 (z − w)ez /3+w /3−xz−yw I exp,n(x, y) −−−→2KGSE(x, y) = dz dw , 12 n→∞ 12 Cπ/3 Cπ/3 + 1 1 2z(z w) 3 3 (z − w)ez /3+w /3−xz−yw I exp,n(x, y) −−−→4KGSE(x, y) = dz dw . 22 n→∞ 22 Cπ/3 Cπ/3 + 1 1 z w

exp,n PROOF. Let us provide a detailed proof for the asymptotics of I11 .Thear- exp,n exp,n guments are almost identical for I12 and I22 , the only difference being the presence of poles that are harmless for the contour deformation that we will perform (see the end of the proof). We use Laplace’s method on the contour integrals in (65). It is clear that the asymptotic behavior of the integrand is dominated by the term exp[n(f (z) + ] [ ] Cπ/3 f(w)) . Let us study the behavior of Re f(z) along the contour 0 .

Cπ/3 16 [ ] LEMMA 5.9. The contour 0 is steep descent for Re f in the sense that the functions t → Re[f(teiπ/3)] and t → Re[f(te−iπ/3)] are strictly decreasing for t>0. Moreover, for t>1, Re[f(te±iπ/3)] < −2t + 2.

PROOF.Wehave − − 2t Re f teiπ/3 = Re f te iπ/3 =−2t + tanh 1 , 1 + 4t2

16In general, we say that a contour C is steep descent for a real valued function g if g attains a unique maximum on the contour and is monotone along both parts of the contours separated by this maximum. 3070 BAIK, BARRAQUAND, CORWIN AND SUIDAN

Clocal FIG.8. Contours used in the proof of Proposition 5.8. Left: the contour 0∈/ . Right: the contour C0∈/ . so that 2 4 d ± 16(t + 2t ) Re f te iπ/3 =− , dt 1 + 4t2 + 16t4 which is always negative, and less than −2fort>1.

We would like to deform the integration contour in (65) to be the steep-descent Cπ/3 contour 0 . This is not possible due to the pole at 0, and hence we modify the Cπ/3 −1/3 C contour 0 in a n -neighborhood of 0 in order to avoid the pole. We call 0∈/ the resulting contour, which is depicted in Figure 8. For a ﬁxed but large enough n, the integrand in (65) has exponential decay along the tails of the integration contour: this is because for large n, the behavior is governed by enf (z )+nf (w ) and we have that Re[f(z)] −→ − ∞ for z →∞eiπ/3. |z|

Hence, we are allowed to deform the contour to be C0∈/ and write z − w (2z + 2α − 1)(2w + 2α − 1) I exp,n(x, y) = dz dw 11 + − 2 C0∈/ C0∈/ 4zw(z w) (2α 1) (69) × exp n f(z)+ f(w) − n1/3σ(xz+ yw) . We recall that x and y here can be negative. We will estimate (69) by cutting it into two parts: The contribution of the integrations in a neighborhood of 0 will PFAFFIAN SCHUR PROCESSES AND LPP IN A HALF-QUADRANT 3071

converge to the desired limit, and the integrations along the rest of the contour will go to 0 as n goes to infinity. Clocal C Let us denote 0∈/ the intersection of the contour 0∈/ with a ball of radius Ctail := C \ Clocal ε centered at 0, and set 0∈/ 0∈/ 0∈/ . We first show that the RHS of (69) Ctail →∞ with integrations over 0∈/ goes to zero as n for any fixed ε>0. Using Lemma 5.9, we know that for any fixed ε>0, there exists a constant c>0such that for t>ε, Re[f(te±iπ/3)] < −ct. Moreover, using both Taylor approximation (67) and Lemma 5.9, we know that for any z ∈ C0∈/, Re[nf (z)] is bounded uniformly in n on the circular part of C0∈/. This implies that there exists constants ∈ Ctail n0,C0 > 0 such that for n>n0 and z, w 0∈/ we have 1/3 −cn|z| exp n f(z)+ f(w) − n σ(xz+ yw)

σ3 σ3 ˜ −˜ z˜3+ w˜ 3−σxz˜−σyw˜ −1/3 (z w)e 3 3 E2 = n 1/3 local 1/3 local ˜ ˜ ˜ +˜ n C ∈ n C ∈ 4zw(z w) (72) 0/ 0/ 4(z˜ +˜w) × dz˜ dw.˜ 2α − 1 Notice that in the ﬁrst integral in (71),

| −1/3O ˜4 −1/3O ˜ 4 | | −1/3 ˜O ˜3 + −1/3 ˜ O ˜ 3 | |O ˜3 +O ˜ 3 | e n (z )n (w ) = e n z (z ) n w (w )

Hence, choosing ε smaller than σ 3/3, the integrand in (71) has exponential decay 1/3Clocal along the tails of the contour n 0∈/ , so that by dominated convergence, E1 goes to 0 as n goes to infinity. It is also clear that E2 goes to 0 as n goes to infinity. 1/3Clocal Finally, let us explain why the contour n 0∈/ in (70) can be replaced by Cπ/3 ± 1 . Using the exponential decay of the integrand for z, w in direction π/3, 1/3Clocal we can extend n 0∈/ to an infinite contour. Then, using Cauchy’s theorem and again the exponential decay, this infinite contour can be freely deformed (without Cπ/3 ˜ = crossing any pole) to 1 . We obtain, after a change of variables σz z and likewise for w, 1 z − w 3 + 3 − − I exp,n(x, y) −−−→ dz dw ez /3 w /3 xz yw. 11 n→∞ 2 Cπ/3 Cπ/3 + (2iπ) 1 1 4zw(z w) exp,n exp,n When adapting the same method for I12 and I22 , we could in principle encounter poles of 1/(2α − 1 − 2z) or 1/(2α − 1 − 2w). However, assuming that α>1/2, these poles are in the positive real part region, and given the definitions of contours used in Section 4.3, we can always perform the above contour deformations.

exp,n Now, we check that the kernel R22 satisﬁes the hypothesis (3) of Proposi- tion 5.7.

PROPOSITION 5.10. Assume that f(x,y) is a smooth function such that all its partial derivatives have exponential decay at inﬁnity. Then exp,n R f(x,y)dx dy − δ (x, y)f (x, y) dx dy −−−→ 0. 22 n→∞

exp,n PROOF. It is enough to prove the proposition replacing R22 by | − | n2sgn(y − x)e x y n.

We can write | − | n2sgn(y − x)e x y nf(x,y)dx dy | − | = f(x,y) − f(x,x) n2e x y n y>x | − | − f(y,x) − f(x,x) n2e x y n. y>x We make the change of variables y = x + z/n, and use the Taylor approximations 2 3 z z z − f(y,x) − f(x,x) = ∂ f(x,x) + ∂2f(x,x) + ∂3f(x,x) + o z3n 3 , n x 2n2 x 6n3 x 2 3 z z z − f(x,y) − f(x,x) = ∂ f(x,x) + ∂2f(x,x) + ∂3f(x,x) + o z3n 3 . n y 2n2 y 6n3 y PFAFFIAN SCHUR PROCESSES AND LPP IN A HALF-QUADRANT 3073

The quadratic terms will cancel, so that | − | n2sgn(y − x)e x y nf(x,y)dx dy −z = ∂yf(x,x) − ∂xf(x,x) dx ze dz 1 − − + ∂3f(x,x) − ∂3f(x,x) dx z3e z dz + o n 2 n2 y x so that | − | n2sgn(y − x)e x y nf(x,y)dx dy C −2 = ∂yf(x,x) − ∂xf(x,x) dx + + o n . n2

5.3. Conclusion. Wehavetoprovethatwhenα>1/2, 4/3 1/3 GSE (73) lim P H(n, n) < 4n + 2 n z = Pf J − K 2 . n→∞ L (z,∞) Since P + 4/3 1/3 = − exp,n H(n, n) < 4n 2 n z Pf J K L2(z,∞) we are left with checking that the kernel Kexp,n satisfies the assumptions in Proposition 5.7: Hypothesis (3) is proved by Proposition 5.10. Hypothesis (2) exp GSE is proved by Proposition 5.8 (with Bn being K ). Note that K (x, y) (defined in Lemma 2.7) does not have exponential decay in all directions, but since we compute the Fredholm Pfaffian on L2(z, ∞), we can replace KGSE(x, y) by GSE 1x,y≥zK (x, y) so that exponential decay holds in all directions. Hypothesis (1) is proved by the following.

LEMMA 5.11. Assume α>1/2 and let z ∈ R be ﬁxed. There exist a constant c0 > 0 and for all r, s ∈ Z≥0, there exist positive constants Cr,s ,n0 such that for n>n0 and x,y > z, r s exp,n −c0x−c0y ∂x ∂yI11 (x, y) 0 such that exp,n −c0x−c0y exp,n −c0x−c0y I11 (x, y)

exp,n PROOF. We start with the expression for I11 in (69). We have already seen in the proof of Proposition 5.8 that using both Taylor approximation (67) 3074 BAIK, BARRAQUAND, CORWIN AND SUIDAN and Lemma 5.9, Re[nf (z)] is bounded by some constant Cf uniformly in n for z ∈ C0∈/. Hence, there exists a constant Cf such that (z − w)(2z + 2α − 1)(2w + 2α − 1) − 1/3 + < dz dw e2Cf n σ(xz yw). (69) + C0∈/ C0∈/ 4zw(z w) Now make the change of variables z˜ = n1/3z and likewise for w,sothat z˜ −˜w (69) < dz˜ dw˜ 1/3 1/3 ˜ ˜ ˜ +˜ n C0∈/ n C0∈/ 4zw(z w) − − − ˜+ ˜ × 2n 1/3z˜ + 2α − 1 2n 1/3w˜ + 2α − 1 e2Cf σ(xz yw). 1/3 Since the real part of z and w stays above 1/2 along the contour n C0∈/,weget that for x,y > 0 (69) 0. Moreover, the estimates used in Proposition 5.8 show that for x and y in a compact subset, the kernel is bounded uniformly in n, so that there exists C00 > 0 such that for all x,y > z, exp,n − − I11 (x, y) 0 such that r s exp,n − − ∂x ∂yI11 (x, y)

6. Asymptotic analysis in other regimes: GUE, GOE, crossovers. In this section, we prove GUE and GOE and Gaussian asympotics and discuss the phase transition. In these regimes, the phenomenon of coalescence of points in the limiting point process—which introduces subtleties in the asymptotic analysis of Sec- tion 5—is not present, and the proofs follow a more standard approach.

6.1. Phase transition.Whenwemakeα vary in (0, ∞), the fluctuations of H(n, n) obey a phase transition already observed in [6] in the geometric case. Let us explain why such a transition occurs. When α goes to infinity, the weights on the diagonal go to 0. It is then clear that the last passage path to (n, n) will avoid the diagonal. The total passage time will be the passage time from (1, 0) to (n, n − 1), which has the same law as H(n− 1,n− 1) when α = 1. Thus, the fluctuations should be the same when α equal 1 or infinity, and by interpolation, for any α ∈ (1, +∞). The first part of Theorem 1.4 shows that this is in fact the case for α ∈ (1/2, ∞).Whenα is very small however, we expect that the last passage path will stay close to the diagonal in order to collect most of the large weights on the diagonal. Hence, the last passage time will be the sum of O(n) random PFAFFIAN SCHUR PROCESSES AND LPP IN A HALF-QUADRANT 3075

variables, and hence fluctuate on the n1/2 scale with Gaussian fluctuations. The critical value of α separating these two very different fluctuation regimes occurs at 1/2. Let us give an argument showing why the critical value is 1/2, using a symmetry of the Pfaffian Schur processes.

LEMMA 6.1. The half-space last passage time H(n, n) when the weights are E(α) on the diagonal and E(1) away from the diagonal, has the same distribution as the half-space last passage time H(n,˜ n) when the weights are E(α) on the ﬁrst row and E(1) everywhere else.

PROOF. Apply Proposition 3.4 to the Pfafﬁan Schur process with single variable specializations considered in Proposition 3.10,andtakeq → 1.

Let us study where the transition should occur in the latter H˜ model. We have the equality in law % x ˜ (d) ¯ (74) H(n, n) = max w1i + H(n− 1,n− x) , x∈Z >0 i=1 where H(n¯ − 1,n− x) is the half-space last passage time from point (n, n) to point (x, 2). It is independent of the w1i , and has the same law as H(n− 1,n− x) when α = 1. For n going to infinity with x = (1 − κ)n, κ ∈[0, 1], we know from ¯ Theorem 1.4 (which√ is independent from the present discussion) that H(n− 1,n− x)/n goes to (1 + κ)2. Hence, ! H˜ (n, n) 1 − κ √ lim = max + (1 + κ)2 n→∞ n κ∈[0,1] α (75) ⎧ ⎨⎪4ifα ∈ (1/2, 1), = 1 ⎩⎪ if α ≤ 1/2. α(1 − α) This suggests that the transition happens when α = 1/2. The fact that the maximum in (75) is attained for κ = (α/(1 − α))2 when α<1/2 even suggests that − the fluctuations are Gaussian with variance 1 2α on the scale n1/2. The fluctu- α2(1−α)2 x ¯ − − ations of i=1 w1i are dominant compared to those of H(n x,n 1). We do not attempt to make this argument complete—it would require proving some concen- tration bounds for H(n¯ − 1,n− x) when x = (1 − κ)n+ o(n)—but we provide a computational derivation of the Gaussian asymptotics in Section 6.4. See also [3, 10] for more details on similar probabilistic arguments. In the case α = 1/2, the second part of Theorem 1.3 shows that the fluctuations of H(n, n) are on the scale n1/3 and Tracy–Widom GOE distributed. It suggests that the value of x that maximizes (74) is close to 0 on the scale at most n2/3. 3076 BAIK, BARRAQUAND, CORWIN AND SUIDAN

6.2. GOE asymptotics. In this section, we prove the second part of Theo- rem 1.3. We consider a scaling of the kernel slightly different than what we used in Section 5.2.Wesetσ = 24/3 as before, r(x, y) as in (64), and here σ 2n2/3Kexp r(x, y) σn1/3Kexp r(x, y) exp,n := 11 12 K (x, y) 1/3 exp exp . σn K21 r(x, y) K22 r(x, y) We write Kexp,n(x, y) = I exp,n(x, y) + Rexp,n(x, y), according to the decomposition made in Section 4.4. The pointwise asymptotics of Proposition 5.8 can be readily adapted when α = 1/2.

PROPOSITION 6.2. For α = 1/2, we have z − w 3 + 3 − − I exp,n(x, y) −−−→ dz dw ez /3 w /3 xz yw, 11 n→∞ Cπ/3 Cπ/3 z + w 1 1 w − z 3 + 3 − − I exp,n(x, y) −−−→ dz dw ez /3 w /3 xz yw, 12 n→∞ Cπ/3 Cπ/3 + 1 −1/2 2w(z w) z − w 3 + 3 − − I exp,n(x, y) −−−→ dz dw ez /3 w /3 xz yw. 22 n→∞ Cπ/3 Cπ/3 + 1 1 4zw(z w) exp,n PROOF. We adapt the asymptotic analysis from Proposition 5.8.ForI11 , exp,n the arguments are strictly identical. For I12 , we have a pole for the variable w 1 in 0 [coming from the factor 2α−1−2w in (68)] which must stay on the right of C exp,n the contour. Hence, we cannot use the contour 0∈/ as for I11 but another mod- Cπ/3 −1/3 ification of the contour 0 in a n -neighborhood of 0 such that the contour passes to the left of 0. We denote by C0∈ this new contour; see Figure 9. Instead of working with a formula similar to (69), we start with z − w 2α − 1 + 2z I exp,n(x, y) = σn1/3 dz dw 12 + − − C0∈ C0∈ 2z(z w) 2α 1 2w (76) × exp n f(z)+ f(w) − n1/3σ(zx+ wy) , where f is defined in (66). The arguments used to perform the asymptotic analysis of (69) can be readily adapted to (76). Cπ/3 This explains the choice of −1/2 as the contour for w in I12 in the limit. For exp,n I22 however, thanks to the deformation of contours performed in Section 4.3, the presence of poles for z and w at 0 has no influence in the computation of the limit, and the asymptotic analysis is very similar as in Proposition 5.8. = ¯exp,n = PROPOSITION 6.3. Assume α 1/2. Then we have R11 (x, y) R¯exp,n(x, y) = R¯exp,n(x, y) = 0, and for any x,y ∈ R, 12 21 3 − dz 3 − dz sgn(x − y) R¯exp,n(x, y) −−−→ − ez /3 zx + ez /3 zy − . 22 n→∞ Cπ/3 Cπ/3 1 4z 1 4z 4 PFAFFIAN SCHUR PROCESSES AND LPP IN A HALF-QUADRANT 3077

FIG.9. The contour C0∈ used in the end of the proof of Proposition 6.2, for the asymptotics of exp K12 .

¯exp,n = ¯exp,n = ¯exp,n = PROOF. The fact that R11 (x, y) R12 (x, y) R21 (x, y) 0isjust ¯exp,n ¯exp,n by deﬁnition of R in Section 4.4. Regarding R22 ,whenx>y,

1/3 1/3 enf (z )−xσn z enf (z )−yσn z R¯exp,n(x; y) =− dz + dz 22 Cπ/3 Cπ/3 1/4 4z 1/4 4z (77) 1 − 1/3| − | 1 + e σn x y z dz − , Cπ/3 1/4 2z 4 ¯exp ; where f is deﬁned in the proof of Proposition 5.8,andR22 (i, x j,y) is antisymmetric. Similar arguments as in Proposition 5.8 show that

nf (z )−xσn1/3z e 3 − dz dz −−−→ ez /3 zx , Cπ/3 n→∞ Cπ/3 1/4 4z 1 4z ¯ so that the proposition is proved, using the antisymmetry of R22.

At this point, we have shown that when α = 1/2, exp,n k GOE k Pf K (xi,xj ) = −−−→ Pf K (xi,xj ) = . i,j 1 q→1 i,j 1 In order to conclude that the Fredholm Pfafﬁan likewise has the desired limit, we need a control on the entries of the kernel Kexp,n, in order to apply dominated convergence. 3078 BAIK, BARRAQUAND, CORWIN AND SUIDAN

LEMMA 6.4. Assume α = 1/2 and let a ∈ R be ﬁxed. There exist positive constants C,c ,mfor n>mand x,y > a, 1 2/3 exp,n −c1x−c1y 1/3 exp,n −c1x n K11 (x, y)

PROOF. The proof is very similar to that of Lemma 5.11. The contour for w exp C C in K12 used in the asymptotic analysis is now 0∈ instead of 0∈/, resulting in an →∞ exp,n absence of decay as y .ThekernelI22 has exponential decay in x and y exp,n using arguments similar with Lemma 5.11. However, the bound on K22 is only constant, due to the term sgn(x − y).

The bounds from Lemma 6.4 are such that the hypotheses in Lemma 2.5 are satisfied, and we conclude, applying dominated convergence in the Pfaffian series expansion, that 4/3 1/3 GOE lim P H(n, n) < 4n + 2 n x = Pf J − K 2 . n→∞ L (x,∞) 6.3. GUE asymptotics. In this section, we prove Theorem 1.4. Fix a parameter κ ∈ (0, 1) and assume that m = κn. Again we expect that H(n,m)/nconverges to some constant h depending on κ, with fluctuations on the n1/3 scale. The kernel Kexp can be rewritten as 11 z − w Kexp,n r(x, y) = dz dw 11 Cπ/3 Cπ/3 + 1/4 1/4 4zw(z w) (78) × (2z + 2α − 1)(2w + 2α − 1) 1/3 × exp n fκ (z) + fκ (w) − n σ(xz+ yw) , where

(79) fκ (z) =−hz + log(1 + 2z) − κ log(1 − 2z). It is convenient to parametrize the constant κ by a constant θ ∈ (0, 1/2),andset the values of h and σ such that 1 − 2θ 2 4 8 8κ 1/3 κ = ,h= ,σ= + . 1 + 2θ (1 + 2θ)2 (1 + 2θ)3 (1 − 2θ)3 = = With this choice, fκ (θ) fκ (θ) 0 and by Taylor expansion, σ 3 (80) f (z) = f (θ) + (z − θ)3 + O (z − θ)4 . κ κ 3 The kernel Kexp can be rewritten as 22 ⎧ ⎪ exp ⎨R22 when α>1/2, Kexp = I exp + Rˆexp when α<1/2, 22 22 ⎪ 22 ⎩ ¯exp = R22 when α 1/2, PFAFFIAN SCHUR PROCESSES AND LPP IN A HALF-QUADRANT 3079

where z − w 1 1 I exp r(x, y) = dz dw 22 Cπ/3 Cπ/3 + − − − − 1/4 1/4 z w 2α 1 2z 2α 1 2w 1/3 × exp n gκ (z) + gκ (w) − n σ(xz+ yw) , with gκ (z) =−fκ (−z).Whenα>1/2, − −| − | 1/3 en(gκ (z) fκ (z)) x y n σz Rexp r(x, y) =−sgn(x − y) 2z dz. 22 Cπ/3 − − − + 0 (2α 1 2z)(2α 1 2z) When α<1/2, − −| − | 1/3 en(gκ (z) fκ (z)) x y n σz Rˆexp r(x, y) =−sgn(x − y) 2z dz 22 Cπ/3 − − − + 0 (2α 1 2z)(2α 1 2z) − − 1 2α σn1/3y n(g (z)+g ( 1 2α ))−xσn1/3z dz − e 2 e κ κ 2 Cπ/3 − − 0 2α 1 2z − − 1 2α σn1/3x n(g (z)+g ( 1 2α ))−yσn1/3z dz + e 2 e κ κ 2 Cπ/3 − − 0 2α 1 2z when α = 1/2, − −| − | 1/3 dz R¯exp r(x, y) = sgn(x − y) en(gκ (z) fκ (z)) x y n σz 22 Cπ/3 1/4 2z − n(g (z)+g ( 1 2α ))−xσn1/3z dz + e κ κ 2 Cπ/3 1/4 2z − − n(g (z)+g ( 1 2α ))−yσn1/3z dz sgn(x y) − e κ κ 2 − . Cπ/3 1/4 2z 4 − = − = − = − Since gκ ( θ) gκ ( θ) 0, and gκ ( θ) fκ (θ), Taylor expansion around θ yields σ 3 g (z) = g (−θ)+ (z + θ)3 + O (z + θ)4 . κ κ 3 Scale the kernel Kexp as ⎛ ⎞ σn1/3Kexp(r(x, y)) ⎜ 11 1/3 exp ⎟ ⎜ − 1/3 + σn K12 r(x, y) ⎟ exp,n := ⎜e2nf κ (θ) σn (x y)θ ⎟ K (x, y) ⎝ 1/3 exp ⎠ . 1/3 exp σn K22 (r(x, y)) σn K21 r(x, y) − + 1/3 + e2ngκ ( θ) σn (x y)θ By a conjugation of the kernel preserving the Pfafﬁan, the factor e2nf κ (θ) in exp − exp 2ngκ ( θ) =− − K11 cancels with e in K22 [since fκ (θ) gκ ( θ)], and the factor 3080 BAIK, BARRAQUAND, CORWIN AND SUIDAN

−σn1/3(xθ+yθ) exp −σn1/3(−xθ−yθ) exp e in K11 cancels with e in K22 . This implies that by a change of variables in the Pfafﬁan series expansion, H(n,κn)− hn P ≤ y = Pf J − Kexp,n . σn1/3 L2(y,∞)

PROPOSITION 6.5. We have that 2 2/3 exp,n σ n K11 (x, y) − 3 3 n→∞ z w 2 z + w −xz−yw −−−→ dz dw (2θ + 2α − 1) e 3 3 , Cπ/4 Cπ/4 3 1 1 8θ so that in particular Kexp,n(x, y) −−−→ 0. 11 n→∞

PROOF. We start with (78). Since for ﬁxed n, the integrand has exponential decay in the direction eiφ for any φ ∈ (−π/2,π/2),weareallowedtodeformthe Cπ/3 Cπ/4 Cπ/4 Cπ/4 contour from 1/4 to 1/4 . Then we deform the contour from 1/4 to θ .Thisis valid as soon as we do not cross any pole during the deformation, which is the case ∈ Cπ/4 when θ (0, 1/2). The following shows that the contour θ is steep-descent.

LEMMA 6.6. For any θ ∈ (0, 1/2), the functions t → Re[fκ (θ + t(1 + i))] and t → Re[fκ (θ + t(1 − i))] are strictly decreasing for t>0. Moreover, for d [ + + ] t>1, dt Re fκ (θ t(1 i)) is uniformly bounded away from zero.

PROOF. Straightforward calculations show that d Re f θ + t(1 + i) dt κ 1 =−h(θ + t)+ log (1 + 2θ + 2t)2 + (2t)2 2 κ − log (1 − 2θ − 2t)2 + (2t)2 2 is strictly negative for t>0andθ ∈ (0, 1/2). Calculations are the same for the other branch (in direction 1 − i) of the contour.

Using similar arguments as in Section 5.2, we estimate the integral in two parts. Clocal Cπ/4 We call θ the intersection between θ with a ball of radius ε centred at θ, and we write Cπ/4 = Clocal Ctail θ θ θ . Using the steep-descent properties of Lemma 6.6, we can show that the integration Ctail over θ goes to 0 exponentially fast as n goes to inﬁnity. We are left with ﬁnding PFAFFIAN SCHUR PROCESSES AND LPP IN A HALF-QUADRANT 3081

the asymptotic behavior of (z − w)(2z + 2α − 1)(2w + 2α − 1) Clocal Clocal + θ θ 4zw(z w) + − 1/3 + × en(fκ (z) fκ (w)) n σ(xz yw) dz dw.

We make the change of variables z = θ +˜zn−1/3 and w = θ +˜wn−1/3,andﬁnd

−1/3 − n (z˜ −˜w) n 2/3 (2θ + 2α − 1)(2θ + 2α − 1) 1/3Clocal 1/3Clocal 3 n θ n θ 8θ σ 3 × exp 2nf (θ ) − σn1/3(x + y) exp z˜3 − σ(xz˜ + yw)˜ dz˜ dw,˜ 3 where the n−2/3 factor is due to the Jacobian of the change of variables. Finally, 1/3Clocal we can extend n θ to an inﬁnite contour making a negligible error and then Cπ/4 freely deform the contour to 1 .

exp,n − − For K22 , the presence of the pole√ in 1/(2α 1 2z) imposes the condition α> / − θ α> √κ 1 2 , or equivalently, 1+ κ .

√ κ ROPOSITION For α> √ we have that P 6.7. 1+ κ ,

3 3 z + w −xz−yw z − w e 3 3 σ 2n2/3Kexp,n(x, y) −−−→ dz dw , 22 n→∞ Cπ/4 Cπ/4 − − + 2 0 0 2θ (2α 1 2θ) so that in particular Kexp,n(x, y) −−−→ 0. 22 n→∞

PROOF.Sincen>m= κn, all contours can be deformed to the vertical con- Cπ/2 Cπ/2 ˆ ¯ tours 0 or 1/4 in the expressions for I22,R22, R22 and R22. This is because the quantity egκ (z) has enough decay along vertical contours as long as κ<1andn is large enough. By reversing the deformations of contours performed in Section 4.3,

+ − 1/3 + z − w en(gκ (z) gκ (w)) n σ(xz yw) Kexp r(x, y) = dz dw , 22 Cπ/2 Cπ/2 + − − − − az aw z w (2α 1 2z)(2α 1 2w)

where az,aw are chosen as any value in R<0 when α ≥ 1/2 and any value in 2α−1 ∈ 2α−1 ( 2 , 0) when α<1/2.Sincewehaveassumedthatθ ( 2 , 0), we can freely deform the contours to the contour C|η depicted on Figure 10, where the constant η>0 can be chosen as small as we want. Now, we estimate the behavior of Re[gκ ] on the contour C|η. 3082 BAIK, BARRAQUAND, CORWIN AND SUIDAN

FIG. 10. The contour C|η used in the proof of Proposition 6.7.

LEMMA 6.8. For any θ ∈ (0, 1/2), the functions t → Re[fκ (θ − t(1 + i))] and t → Re[fκ (θ − t(1 − i))] are strictly increasing for t>0. Moreover, for d [ − + ] t>1, dt Re fκ (θ t(1 i)) is uniformly bounded away from zero. In particular, Cπ/4 [ ] the contour −θ is steep descent for Re gκ .

PROOF. The calculations are analogous to the proof of Lemma 6.6.

LEMMA 6.9. For η>0 small enough, the function Re[gκ (−η + iy)] is decreasing for y ∈ (c(η), +∞) where 0

PROOF.Wehave d (κ − 1)(32y3 + (1 + 2η)2) + 8η Re g (−η + iy) = , dy κ 2((1 − 2η)2 + 4y2)((1 + 2η)2 + 4y2) from which the result follows readily because κ<1.

Lemmas 6.8 and 6.9 together imply that for η small enough, the contour C|η is steep descent for the function Re[gκ ]. Then an adaptation of the proof of Proposi- tion 6.5 concludes the proof.

exp =−˜ TurningtothekernelK12 , after the change of variables w w,wehave σ −1n−1/3 z +˜w 2α − 1 + 2z Kexp,n(x, y) = dz dw˜ 12 2 Cπ/3 C2π/3 −˜ − + ˜ (2iπ) 1/4 −1/4 2z(z w) 2α 1 2w (81) 1/3 × exp n fκ (z) − fκ (w)˜ − n σ(z−˜w) , where fκ is deﬁned in (79). PFAFFIAN SCHUR PROCESSES AND LPP IN A HALF-QUADRANT 3083

Clocal Dlocal FIG. 11. Contours used in the proof of Proposition 5.8. Left: the contours θ/∈ and θ/∈ . Right: the contours Cθ/∈ and Dθ/∈.

√ κ ROPOSITION For α> √ P 6.10. 1+ κ , exp,n K (x, y) −−−→ KAi(x, y), 12 n→∞

where KAi is the Airy kernel deﬁned in (1).

Cπ/4 PROOF. We have already seen in Lemma 6.6 that the contour θ is [ ] C3π/4 steep-descent for Re fκ (z) , and in Lemma 6.8 that θ is steep descent for −Re[fκ (z)]. Due to the term 1/(z − w) in (81), we cannot in principle use simul- Cπ/4 C3π/4 taneously the contour θ for z and θ for w. Hence, we deform the contours −1/3 in a n -neighborhood of θ. Let us denote Cθ/∈ and Dθ/∈ the modiﬁed contours; Clocal Dlocal see Figure 11. As in the proof of Proposition 5.8, we call θ/∈ and θ/∈ the intersection of Cθ/∈ and Dθ/∈ with a ball of radius ε around θ, as in Section 5.2.Using the same arguments as in Section 5.2, we can ﬁrst deform the contours in (81)to C D Clocal θ/∈ and θ/∈, and then approximate the integrals by integrations over θ/∈ and Dlocal = +˜ −1/3 = +˜ −1/3 θ/∈ . By making the change of variables z θ zn and w θ wn , and approximating the integrand using the Taylor approximation (80) and straightforward pointwise limits, we are left with 3 3 σ z˜3− σ w˜ 3−xσz˜+yσw˜ −2/3 ˜ ˜ e 3 3 n dz dw − , Cπ/4 C3π/4 1/3 ˜ −˜ 1/4 −1/4 n (z w) 1/3 Clocal − as desired. Note that the integration contours should be n ( 0∈/ θ) and 1/3 Dlocal − Cπ/4 C3π/4 n ( 0∈/ θ), but in the large n limit, we can use 1 and −1 instead for the same reasons as in Proposition 5.8, so that we recognize the Airy kernel. 3084 BAIK, BARRAQUAND, CORWIN AND SUIDAN

Let us denote the pointwise limit of Kexp,n by 0 K (x, y) KGUE(x, y) := Ai . −KAi(y, x) 0 The pointwise convergence of the kernel (Propositions 6.5, 6.7 and 6.10) along with the fact that Kexp,n satisﬁes uniformly the hypotheses of Lemma 2.5 (this is exp,n exp,n exp,n clear for K11 and K22 anditisprovedasinLemma6.4 for K12 ), shows by dominated convergence that √ 2 1/3 GUE lim P H(n,κn)<(1 + κ) n + σn x = Pf J − K 2 . n→∞ L (x,∞) Finally, − GUE = + GUE Pf J K L2(x,∞) det I JK L2(x,∞) I 0 K 0 = det − Ai 0 I 0 KAi L2(x,∞) = − = det(I KAi)L2(x,∞) FGUE(x).

REMARK 6.11. We have assumed in the statement of Theorem 1.4 that m = κn for some ﬁxed κ. A stronger statement holds. When m = κn+ sn2/3−ε for any s ∈ R and ε>0, √ H(n,κn)− (1 + κ)2n − 2sn2/3−ε lim P

2/3−ε 2/3−ε This change results in an additional factor e2sn w−sn log(1−2w) in (78). After the change of variables w = n−1/3w˜ in the proof of Propositions 6.5, 6.7 and 6.10, −ε the additional factor becomes e2n w˜ and does not contribute to the limit.

6.4. Gaussian asymptotics. In this section, we prove the third part of Theo- 1 1 rem 1.3. Assume that α<1/2. The factors 2α−1−2w and 2α−1−2z in the expres- exp exp sions of K12 and K22 in Proposition 1.6 prevent us from deforming the contours to go through the critical point at zero as in Section 5.2. We already know that the LLN of H(n, n) will be different, and so too will the critical point (its position will coincide with the poles above-mentioned). We have already argued in Section 6.1 that H(n, n) 1 −−−→ =: h(α), n n→∞ α(1 − α) 1/2 and we expect Gaussian ﬂuctuations on the n scale. Let rα(x, y) = (nhα + 1/2 1/2 n σαx,nhα + n σαy) for some constants hα,σα depending on α.Thekernel PFAFFIAN SCHUR PROCESSES AND LPP IN A HALF-QUADRANT 3085

Kexp can be rewritten as 11 exp K rα(x, y) 11 z − w (82) = dz dw Cπ/3 Cπ/3 + 1/4 1/4 4zw(z w) + − 1/2 + × (2z + 2α − 1)(2w + 2α − 1)en(fα(z) fα(w)) n σ(xz yw), where

fα(z) =−hαz + log(1 + 2z) − log(1 − 2z). 2α−1 = 1−2α = = 1−2α There are two critical points, fα( 2 ) fα( 2 ) 0. Setting θ 2 ,and − σ > 0 such that σ 2 = f (θ) = 1 2α , Taylor expansions of f around θ and α α α α2(1−α)2 α −θ yield σ 2 f (z) = f (θ) + α (z − θ)2 + O (z − θ)3 , α α 2 σ 2 f (z) = f (−θ)− α (z + θ)2 + O (z + θ)3 . α α 2 Let us rescale the kernel by setting ⎛ ⎞ σ 2nKexp(r (x, y)) ⎜ α 11 α 1/2 exp ⎟ ⎜ − 1/2 + σαn K12 rα(x, y) ⎟ exp,n := ⎜ e2nf α(θ) σn (x y)θ ⎟ K (x, y) ⎝ exp ⎠ . 1/2 exp K22 (rα(x, y)) σαn K21 rα(x, y) − + 1/2 + e2nf α( θ) σαn (x y)θ We can conjugate the kernel without changing the Pfaffian so that factors 1/3 1/2 exp[2nf α(−θ) − σn (−xθ − yθ)] and exp[2nf α(θ) − σn (x + y)θ] cancels 1/2 out each other. Moreover, a factor σαn will be absorbed by the Jacobian of the change of variables in the Fredholm Pfaffian expansion of Kexp,n,sothat − P H(n,κn) h(α)n ≤ = − exp,n 1/2 y Pf J K L2(y,∞). σαn exp = 1−2α Saddle-point analysis of K11 around the critical point θ 2 shows 3 1/2 exp,n σα n K11 (x, y) (83) z − w 2 + 3 − − −−−→ − dz dw zwez /2 w /3 xz yw. n→∞ Cπ/3 Cπ/3 3 1 1 8θ exp Notice that since α<1/2, θ>0. Since the contours for K11 in Proposition 1.6 must stay in the positive real part region, we choose the critical point θ. The derivation of (83) follows the same steps as in Proposition 6.5.Themostim- Cπ/4 portant step is to find a steep-descent contour. The contour θ used in in Propo- sition 6.5 is not a good choice here, but it is easy to check by direct computation Cπ/3 → [ ] ∈ that θ is steep descent for z Re fα for any value of α (0, 1/2). 3086 BAIK, BARRAQUAND, CORWIN AND SUIDAN exp − = Similarly for K22 , by a saddle-point analysis around the critical point θ 2α−1 exp 2 (the restriction on the contours of K22 in Proposition 1.6 impose now to chose the negative critical point), we get that − 1/2 exp,n z w −z2/2−w2/2−xz−yw σαn K (x, y) −−−→ dz dw e . 22 n→∞ Cπ/6 Cπ/6 − 2 1 1 4θ zw Cπ/6 The angle is chosen as π/6 because the contour −θ is steep descent for the function z → Re[fα], which can be checked again by direct computation. Note that it is less obvious here that we can use vertical contours as in the proof of Proposi- tion 6.7, but it is still possible because the integrand is oscillatory with 1/z decay. exp − In the case of K12 , we use the critical point θ for the variable z and θ for the variable w and obtain 1 z 2 − 2 − − Kexp,n(x, y) −−−→ dz dw ez /2 w /2 xz yw. 12 n→∞ Cπ/3 Cπ/6 + − 2 −1 z w w exp,n We recognize the Hermite kernel in the RHS (see [11]), which yields K12 (x, 2+ 2 1 − x y y) −−−→ √ e 4 . n→∞ 2π Finally, using exponential bounds for the kernel as in Lemmas 5.11 and 6.4,we obtain that H(n, n) − h(α)n lim P

2+ 2 G G G G 1 − x y K (x, y) = K (x, y) = 0,K(x, y) =−K (x, y) = √ e 4 . 11 22 12 21 2π It is clear that KG is of rank one and its Fredholm Pfafﬁan is the cumulative distribution function of the standard Gaussian.

6.5. Convergence of the symplectic-unitary transition to the GSE distribution. We have seen in Section 5 that when α>1/2, Kexp is the correlation kernel of a simple point process which converges to a point process where each point has multiplicity two in the limit. The symplectic-unitary kernel KSU is another example of such kernel in the limit η → 0. The framework that we introduced in Section 5—in particular Proposition 5.7—can be used to prove the following.

= − SU SU := PROPOSITION 6.12. Let FSU(x) Pf(J K )L2(x,∞) where K (x, y) KSU(1,x; 1,y)be the cumulative distribution function of the largest eigenvalue of the symplectic-unitary transition ensemble, depending on a parameter η := η1 ∈ (0, +∞). We have that for all x ∈ R,

FSU(x) −−→ FGSE(x) and FSU(x) −−−−→ FGUE(x). η→0 η→+∞ PFAFFIAN SCHUR PROCESSES AND LPP IN A HALF-QUADRANT 3087

SU PROOF. The limit FSU(x) −−−−→ FGUE(x) is straightforward as K con- η→+∞ verges pointwise to KGUE and we readily check that we can apply dominated convergence in the Fredholm Pfafﬁan expansion. SU Regarding the convergence FSU(x) −−→ FGSE(x), the limit of I (x, y) when η→0 η → 0 is straightforward as well, and we compute that − 2 3 − 2 (y − x)exp( (x y) + 2η − η(x + y)) (y − x)exp( (x y) ) SU = 8√η 3 ∼ √ 8η R22 (x, y) . 4 2πη3/2 η→0 4 2πη3/2 SU Then we readily verify that R22 converges to the distribution δ in the same sense as in Proposition 5.10. Hence, KSU converges as η → 0tothekernelK∞ in the sense of Proposition 5.7, and we conclude that SU GSE Pf J − K L2 ∞ −−→ Pf J − K L2 ∞ . (x, ) η→0 (x, )

Acknowledgments. The authors would like to thank Alexei Borodin and Eric Rains for sharing their insights about Pfafﬁan Schur processes. We are especially grateful to Alexei Borodin for discussions related to the dynamics preserving Pfaf- ﬁan Schur processes and for drawing our attention to the reference [22]. We also thank Tomohiro Sasamoto and Takashi Imamura for discussions regarding their paper [38]. Part of this work was done during the stay of J. Baik, G. Barraquand and I. Corwin at the KITP.

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J. BAIK G. BARRAQUAND DEPARTMENT OF MATHEMATICS I. CORWIN UNIVERSITY OF MICHIGAN DEPARTMENT OF MATHEMATICS 530 CHURCH STREET COLUMBIA UNIVERSITY ANN ARBOR,MICHIGAN 48109 2990 BROADWAY USA NEW YORK,NEW YORK 10027 E-MAIL: [email protected] USA E-MAIL: [email protected] [email protected]

T. SUIDAN E-MAIL: [email protected]